Callogenesis and embryogenesis are integral parts of many tissue culture procedures for genetic manipulation in rice. However, the efficiency of both processes is largely dependent on the media constituent especially the plant growth regulators (PGRs) due to the genotype-dependent nature of in vitro culture protocols. Therefore, this study investigates the effect of two PGRs; 2, 4-dichlorophenoxyacetic acid (2, 4-D) and kinetin (Kin) on callus growth and somatic embryogenesis of an important Malaysian rice cultivar (Oryza sativa L. cv. MARDI Siraj 297). Mature rice seeds explants were inoculated in Murashige & Skoog (MS) medium supplemented with different combinations of 2, 4-D (0 to 3.5 mg/L) and Kin (0 to 0.5 mg/L) to induce callogenesis. Parameters for callus growth such as fresh weight (FW), callus induction frequency (CIF), embryogenic callus frequency (ECF), regeneration frequency (RF), number of plantlets per callus (PPC), callus texture and callus color were observed after 35 days of inoculation. The results show that the maximum callus growth was achieved in MS medium supplemented with combination of 2.0 mg/L 2, 4-D and 0.2 mg/L Kin, represented by the highest FW (211 mg), CIF (95%), ECF (90%), RF (100%) and PPC (22 plantlets); along with friable callus texture. Low concentration of 2, 4-D (0 to 0.5 mg/L) in the presence or absence of Kin promotes root growth instead of callus, while high concentrations (above 3.0 mg/L) retard the callus formation. The embryogenic calli from this optimized PGRs combination were successfully formed shoots in MS medium supplemented with 2 mg/L BAP and 1 mg/L NAA, followed by rooting in PGRs-free MS medium. This finding provides an efficient protocol for callogenesis and somatic embryogenesis of MARDI Siraj 297, since this is the first published report regarding somatic embryogenesis induction of this cultivar.
Citation: Noorhazira Sidek, Rosimah Nulit, Yap Chee Kong, Christina Yong Seok Yien, Rogayah Sekeli, Mariam F. EL-Barghathi. Callogenesis and somatic embryogenesis of Oryza sativa L. (cv. MARDI Siraj 297) under the influence of 2, 4-dichlorophenoxyacetic acid and kinetin[J]. AIMS Agriculture and Food, 2022, 7(3): 536-552. doi: 10.3934/agrfood.2022033
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Callogenesis and embryogenesis are integral parts of many tissue culture procedures for genetic manipulation in rice. However, the efficiency of both processes is largely dependent on the media constituent especially the plant growth regulators (PGRs) due to the genotype-dependent nature of in vitro culture protocols. Therefore, this study investigates the effect of two PGRs; 2, 4-dichlorophenoxyacetic acid (2, 4-D) and kinetin (Kin) on callus growth and somatic embryogenesis of an important Malaysian rice cultivar (Oryza sativa L. cv. MARDI Siraj 297). Mature rice seeds explants were inoculated in Murashige & Skoog (MS) medium supplemented with different combinations of 2, 4-D (0 to 3.5 mg/L) and Kin (0 to 0.5 mg/L) to induce callogenesis. Parameters for callus growth such as fresh weight (FW), callus induction frequency (CIF), embryogenic callus frequency (ECF), regeneration frequency (RF), number of plantlets per callus (PPC), callus texture and callus color were observed after 35 days of inoculation. The results show that the maximum callus growth was achieved in MS medium supplemented with combination of 2.0 mg/L 2, 4-D and 0.2 mg/L Kin, represented by the highest FW (211 mg), CIF (95%), ECF (90%), RF (100%) and PPC (22 plantlets); along with friable callus texture. Low concentration of 2, 4-D (0 to 0.5 mg/L) in the presence or absence of Kin promotes root growth instead of callus, while high concentrations (above 3.0 mg/L) retard the callus formation. The embryogenic calli from this optimized PGRs combination were successfully formed shoots in MS medium supplemented with 2 mg/L BAP and 1 mg/L NAA, followed by rooting in PGRs-free MS medium. This finding provides an efficient protocol for callogenesis and somatic embryogenesis of MARDI Siraj 297, since this is the first published report regarding somatic embryogenesis induction of this cultivar.
The nonlinear Ginzburg-Landau equation plays an important role in the studies of physics, which describes many interesting phenomena and has been studied extensively (see [1] for a more detailed description). The fractional Ginzburg-Landau equation [2,3,4] is employed to describe processes in media with fractional dispersion or long-range interaction. It becomes very popular because the fractional derivative and fractional integral have broad applications in different fields of science [5,6,7,8,9,10].
Our work focuses on the existence of invariant measures of the autonomous fractional stochastic delay Ginzburg-Landau equations on Rn:
du(t)+(1+iν)(−Δ)αu(t)dt+(1+iμ)|u(t)|2βu(t)dt+λu(t)dt=G(x,u(t−ρ))dt,+∞∑k=1(σ1,k(x)+κ(x)σ2,k(u(t)))dWk(t), t>0, | (1.1) |
with initial condition
u(s)=φ(s),s∈[−ρ,0], | (1.2) |
where u(x,t) is a complex-valued function on Rn×[0,+∞). In (1.1), i is the imaginary unit, α,β,μ,ν and λ are real constants with β>0,λ>0 and ρ>0. (−Δ)α with 0<α<1 is the fractional Laplace operator, σ1,k(x)∈L2(Rn) and σ2,k(u):C→R are nonlinear functions, κ(x)∈L2(Rn)⋂L∞(Rn) and {Wk}∞k=1 is a sequence of independent standard real-valued Wiener process on a complete filtered probability space (Ω, F, {Ft}t∈R,P), where {Ft}t∈R is an increasing right continuous family of sub-σ-algebras of F that contains all P-null sets.
The Ginzburg-Landau equation with fractional derivative was first introduced in [2]. There is a large amount of literature which was used for investigating fractional deterministic Ginzburg-Landau equations such as [1] and stochastic equations such as [11,12,13,14,15,16,17]. These papers had respectively researched the long-time deterministic as well as random dynamical systems of fractional equations with autonomous forms and non-autonomous forms. However, in spite of quite a lot of contribution of the works, no result is provided for the existence of pathwise pullback random attractors and invariant measures for the delay stochastic Ginzburg-Landau equations.
The delay differential equations [18] was described the dynamical systems that rely on current and past historical states. For the past few years, researchers had made great progress in the study of linear and nonlinear delay differential equations, see [20,21]. Delay differential equations are widely used in many fields, so investigating the solutions of equations has profound significance. Therefore, it's necessary that we establish the dynamics of delay stochastic Ginzburg-Landau equations.
The goal of this paper is to prove the existence of invariant measures of the stochastic Eqs (1.1) and (1.2) in L2(Ω;C([−ρ,0],L2(Rn))) by applying Krylov-Bogolyubov's method. The main difficulty of this paper is that deducing the uniform estimates of solutions (because of the nonlinear term (1+iμ)|u(t)|2βu(t) and complex-valued solutions), proving the weak compactness of a set distribution laws of the segments of solutions in L2(Ω;C([−ρ,0],L2(Rn))) (because the standard Sobolev embeddings are not compact on unbounded domains Rn), and establishing the equicontinuity of solutions in L2(Ω;C([−ρ,0],L2(Rn))) (because the uniform estimates in L2(Ω;C([−ρ,0],L2(Rn))) are not sufficient, and the uniform estimates in L2(Ω;C([−ρ,0],H1(Rn))) are needed).
For the estimates of the nonlinear term (1+iμ)|u(t)|2βu(t), we apply integrating by parts and nonnegative definite quadratic form. There are Several methods to handle the noncompact on unbounded domain, including weighted spaces [22,23,24], weak Feller approach [25,26] and uniform tail-estimates [23,27]. We first obtain the uniform estimates of the tail of the solution as well as the technique of dyadic division, then establish the weak compactness of a set of probability distribution of solutions in C([−ρ,0],L2(Rn)) applying the Ascoli-Arzelˊa theorem.
Let S be the Schwartz space of rapidly decaying C∞ functions on Rn. The fractional Laplace operator (−Δ)α for 0<α<1 is defined by, for u∈S,
(−Δ)αu(x)=−12C(n,α)∫Rnu(x+y)+u(x−y)−2u(x)|y|n+2αdy, x∈Rn, |
where C(n,α) is a positive constant given by
C(n,α)=α4αΓ(n+2α2)πn2Γ(1−α). |
By [28], the inner product ((−Δ)α2u,(−Δ)α2v) in the complex field is defined by
((−Δ)α2u,(−Δ)α2v)=C(n,α)2∫Rn∫Rn(u(x)−u(y))(ˉv(x)−ˉv(y))|x−y|n+2αdxdy, |
for u∈Hα(Rn). The fractional Sobolev space Hα(Rn) is endowed with the norm
‖u‖2Hα(Rn)=‖u‖2L2(Rn)+2C(n,α)‖(−Δ)α2u‖2L2(Rn). |
About the fractional derivative of fractional Ginzburg-Landau equations, there is another statement in [29].
We organize the article as follows. In Section 2, we establish the well-posedness of (1.1) and (1.2) in L2(Ω;C([−ρ,0],H)). In Sections 3 and 4, we derive the uniform estimates of solutions in L2(Ω;C([−ρ,0],H)) and L2(Ω;C([−ρ,0],V)), respectively. In Section 5, the existence of invariant measures is obtained.
In this section, we show the nonlinear drift term and the diffusion term in (1.1) which are needed for the well-posedness of the stochastic delay Ginzburg-Landau Eqs (1.1) and (1.2) defined on Rn.
We assume that G:Rn×C→C is continuous and satisfies
|G(x,u)|≤|h(x)|+a|u|, ∀x∈Rn, u∈C | (2.1) |
and
|∇G(x,u)|≤|ˆh(x)|+ˆa|∇u|, ∀x∈Rn, u∈C, | (2.2) |
where a and ˆa>0 are constants and h(x),ˆh(x)∈L2(Rn). Moreover, G(x,u) is Lipschitz continuous in u∈C uniformly with respect to x∈Rn. More precisely, there exists a constant CG>0 such that
|G(x,u1)−G(x,u2)|≤CG|u1−u2|, ∀x∈Rn, u1,u2∈C. | (2.3) |
For the diffusion coefficients of noise, we suppose that for each k∈N+
∞∑k=1‖σ1,k‖2<∞, | (2.4) |
and that σ2,k(u):C→R is globally Lipschitz continuous; namely, for every k∈N+, there exists a positive number αk such that for all s1,s2∈C,
|σ2,k(s1)−σ2,k(s2)|≤αk|s1−s2|. | (2.5) |
We further assume that for each k∈N+, there exist positive numbers βk, ˆβk, γk and ˆγk such that
|σ2,k(s)|≤βk+γk|s|, ∀s∈C, | (2.6) |
and
|∇σ2,k(s)|≤ˆβk+ˆγk|∇s|, ∀s∈C, | (2.7) |
where ∞∑k=1(α2k+β2k+γ2k+ˆβ2k+ˆγ2k)<+∞. In this paper, we deal with the stochastic Eqs (1.1) and (1.2) in the space C([−ρ,0],L2(Rn)). In the following discussion, we denote by H=L2(Rn), V=H1(Rn).
A solution of problems (1.1) and (1.2) will be understood in the following sense.
Definition 2.1. We suppose that φ(s)∈L2(Ω,C([−ρ,0],H)) is F0-measurable. Then, a continuous H-valued Ft-adapted stochastic process u(x,t) is named a solution of problems (1.1) and (1.2), if
1) u is pathwise continuous on [0,+∞), and Ft-adapted for all t≥0,
u∈L2(Ω,C([0,T],H))⋂L2(Ω,L2([0,T],V)) |
for all T>0,
2) u(s)=φ(s) for −ρ≤s≤0,
3) For all t≥0 and ξ∈V,
(u(t),ξ)+(1+iν)∫t0((−Δ)α2u(s),(−Δ)α2ξ)ds+∫t0∫Rn(1+iμ)|u(s)|2βu(s)ξ(x)dxds+λ∫t0(u(s),ξ)ds=(φ(0),ξ)+∫t0(G(s,u(s−ρ)),ξ)ds+∞∑k=1∫t0(σ1,k(x)+κ(x)σ2,k(u(s)),ξ)dWk(s), | (2.8) |
for almost all ω∈Ω.
By the Galerkin method and the argument of Theorem 3.1 in [30], one can verify that if (2.1)–(2.7) hold true, then, for every F0-measurable function φ(s)∈L2(Ω,C([−ρ,0],H)), the problems (1.1) and (1.2) has a unique solution u(x,t) in the sense of Definition 2.1.
Now, we establish the Lipschitz continuity of the solutions of the problems (1.1) and (1.2) with respect to the initial data in L2(Ω,C([−ρ,0],H)).
Theorem 2.2. Suppose (2.1)–(2.6) hold, and F0-measurable function φ1,φ2∈L2(Ω,C([−ρ,0],H)). If u1=u(t,φ1) and u2=u(t,φ2) are the solutions of the problems (1.1) and (1.2) with initial data φ1 and φ2, respectively, then, for any t≥0,
E[sup−ρ≤s≤t‖u(s,φ1)−u(s,φ2)‖2]+E[∫t0‖u(s,φ1)−u(s,φ2)‖2Vds] |
≤C1e˜C1tE[sup−ρ≤s≤0‖φ1(s)−φ2(s)]‖2], |
where C1 and ˜C1 are positive constants independent of φ1 and φ2.
Proof. Since both u1 and u2 are the solutions of the problems (1.1) and (1.2), we have, for all t≥0,
u1−u2+(1+iν)∫t0(−Δ)α(u1−u2)ds+(1+iμ)∫t0(|u1|2βu1−|u2|2βu2)ds+λ∫t0(u1−u2)ds=φ1(0)−φ2(0)+∫t0(G(x,u1(s−ρ))−G(x,u2(s−ρ)))ds+∞∑k=1∫t0κ(x)(σ2,k(u1)−σ2,k(u2))dWk. | (2.9) |
By (2.9), the integration by parts of Ito's formula and taking the real parts, we get, for all t≥0,
‖u1−u2‖2+2∫t0‖(−Δ)α2(u1−u2)‖2ds+2Re∫t0∫Rn(ˉu1−ˉu2)[|u1|2βu1−|u2|2βu2]dxds+2λ∫t0‖u1−u2‖2ds=‖φ1(0)−φ2(0)‖2+2Re∫t0(u1−u2,G(x,u1(s−ρ))−G(x,u2(s−ρ)))ds+∞∑k=1∫t0‖κ(x)(σ2,k(u1)−σ2,k(u2))‖2ds+2Re∫t0(u1−u2,∞∑k=1κ(x)(σ2,k(u1)−σ2,k(u2)))dWk(s). | (2.10) |
For the third term in the first row of (2.10), one has
2Re∫t0∫Rn(ˉu1−ˉu2)[|u1|2βu1−|u2|2βu2]dxds=∫t0∫Rn2|u1|2β+2+2|u2|2β+2−2Re(u1ˉu2)(|u1|2β+|u2|2β)dxds≥∫t0∫Rn2|u1|2β+2+2|u2|2β+2−2|u1||u2|(|u1|2β+|u2|2β)dxds≥∫t0∫Rn2|u1|2β+2+2|u2|2β+2−(|u1|2+|u2|2)(|u1|2β+|u2|2β)dxds=∫t0∫Rn|u1|2β+2+|u2|2β+2−|u1|2β|u2|2−|u2|2β|u1|2dxds=∫t0∫Rn(|u1|2β−|u2|2β)(|u1|2−|u2|2)dxds≥0. |
By (2.10), we deduce that for t≥0,
E[sup0≤r≤t‖u1(r)−u2(r)‖2]≤E[sup−ρ≤s≤0‖φ1(s)−φ2(s)‖2]+2E[∫t0‖u1−u2‖⋅‖G(x,u1(s−ρ))−G(x,u2(s−ρ))‖ds] +∞∑k=1E[∫t0‖κ(σ2,k(u1)−σ2,k(u2))‖2ds] +2E[sup0≤r≤t|∞∑k=1∫r0(u1−u2,κ(x)(σ2,k(u1)−σ2,k(u2))dWk(s))|]. | (2.11) |
For the second term on the right-hand side of (2.11), by (2.3), one has
2E[∫t0‖u1−u2‖⋅‖G(x,u1(s−ρ))−G(x,u2(s−ρ))‖ds]≤E[∫t0‖u1−u2‖2ds]+E[∫t0‖G(x,u1(s−ρ))−G(x,u2(s−ρ))‖2ds]≤E[∫t0‖u1−u2‖2ds]+C2GE[∫t0‖u1(s−ρ)−u2(s−ρ)‖2ds]=E[∫t0‖u1−u2‖2ds]+C2GE[∫t−ρ−ρ‖u1−u2‖2ds]≤(1+C2G)E[∫t0‖u1−u2‖2ds]+C2GE[∫0−ρ‖φ1(s)−φ2(s)‖2ds]≤(1+C2G)∫t0E[sup0≤r≤s‖u1−u2‖2]ds+ρC2GE[sup−ρ≤s≤0‖φ1(s)−φ2(s)‖2]. |
For the third term on the right-hand side of (2.11), by (2.5), we have
∞∑k=1E[∫t0‖κ(x)(σ2,k(u1)−σ2,k(u2))‖2ds]≤‖κ(x)‖2L∞∞∑k=1α2kE[∫t0‖u1−u2‖2ds]≤‖κ(x)‖2L∞∞∑k=1α2k∫t0E[sup0≤r≤s‖u1−u2‖2]ds. | (2.12) |
For the forth term on the right-hand side of (2.11), by Burkholder-Davis-Gundy's inequality, one has
2E[sup0≤r≤t|∞∑k=1∫r0(u1−u2,κ(x)(σ2,k(u1)−σ2,k(u2))dWk(s))|]≤B1E[(∫t0∞∑k=1|(u1−u2,κ(x)(σ2,k(u1)−σ2,k(u2)))|2ds)12]≤B1E[(∫t0∞∑k=1‖u1−u2‖2⋅‖κ‖2L∞⋅‖σ2,k(u1)−σ2,k(u2)‖2ds)12]≤B1E[sup0≤s≤t‖u1−u2‖⋅‖κ‖L∞⋅(∞∑k=1α2k)12(∫t0‖u1−u2‖2ds)12]≤12E[sup0≤s≤t‖u1−u2‖2]+12B21‖κ‖2L∞∞∑k=1α2kE[∫t0sup0≤r≤s‖u1−u2‖2ds], | (2.13) |
where B1 is a constant produced by Burkholder-Davis-Gundy's inequality.
It follows from (2.11)–(2.13) that for all t≥0,
E[sup0≤r≤t‖u1(r)−u2(r)‖2]≤2(1+ρC2G)E[sup−ρ≤s≤0‖φ1(s)−φ2(s)‖2]+2[1+C2G+(1+12B21)‖κ‖2L∞∞∑k=1α2k]∫t0E[sup0≤r≤s‖u1(r)−u2(r)‖2]ds. | (2.14) |
Applying Gronwall inequality to (2.14), we obtain that for all t≥0,
E[sup0≤r≤t‖u1(r)−u2(r)‖2]≤2(1+ρC2G)ec1tE[sup−ρ≤s≤0‖φ1(s)−φ2(s)‖2], | (2.15) |
where c1=2[1+C2G+(1+12B21)‖κ‖2L∞∞∑k=1α2k]. By (2.10), there exists c2 such that for all t≥0,
E[∫t0‖u1−u2‖2Vds]≤˜c2ec2tE[sup−ρ≤s≤0‖φ1(s)−φ2(s)‖2]. |
We assume that a, αk and γk are small enough in the sense, there exists a constant p≥2 such that
21−12p(2p−1)2p−12pa+2p(2p−1)‖κ‖2L∞∞∑k=1(α2k+γ2k)<pλ. | (3.1) |
By (3.1), one has
2‖κ‖2L∞∞∑k=1γ2k<λ, | (3.2) |
and
√2a+2‖κ‖2L∞∞∑k=1γ2k<λ. | (3.3) |
The inequalities (3.1)–(3.3) are used to establish the uniform tail-estimate of the solution of (1.1) and (1.2).
Lemma 3.1. Suppose (2.1)–(2.6) and (3.2) hold. If φ(s)∈L2(Ω;C([−ρ,0],H)), then, for all t≥0, there exists a positive constant μ1 such that the solution u of (1.1) and (1.2) satisfies
E[‖u(t)‖2]+∫t0eμ1(s−t)E(‖u(s)‖2V)ds+∫t0eμ1(s−t)E(‖u(s)‖2β+2L2β+2)ds |
≤M1E[sup−ρ≤s≤0‖φ(s)‖2]+~M1, | (3.4) |
and
∫t+ρ0E[‖u(s)‖2V]ds≤(M1(t+ρ)+1+√2aρC(n,α))E[sup−ρ≤s≤0‖φ(s)‖2]+√2(t+ρ)aC(n,α)‖h(x)‖2 |
+2(t+ρ)C(n,α)∞∑k=1(‖σ1,k‖2+2β2k‖κ(x)‖2)+˜M1(t+ρ), |
where ˜M1 is a positive constant independent of φ.
Proof. By (1.1) and the integration by parts of Ito's formula, we have for all t≥0,
‖u(t)‖2+2∫t0‖(−Δ)α2u(s)‖2ds+2∫t0‖u(s)‖2β+2L2β+2ds+2λ∫t0‖u(s)‖2ds=2Re∫t0(u(s),G(x,u(s−ρ)))ds+‖φ(0)‖2+∞∑k=1∫t0‖σ1,k(x)+κ(x)σ2,k(u(s))‖2ds+2Re∫t0(u(s),∞∑k=1σ1,k(x)+κ(x)σ2,k(u(s)))dWk(s). | (3.5) |
The system (3.5) can be rewritten as
d(‖u(t)‖2)+2‖(−Δ)α2u(t)‖2dt+2‖u(t)‖2β+2L2β+2dt+2λ‖u(t)‖2dt=2Re(u(t),G(x,u(t−ρ)))dt+∞∑k=1‖σ1,k(x)+κ(x)σ2,k(u(t))‖2dt+2Re(u(t),∞∑k=1σ1,k(x)+κ(x)σ2,k(u(t)))dWk(t). | (3.6) |
Assume that μ1 is a positive constant, one has
eμ1t‖u(t)‖2+2∫t0eμ1s‖(−Δ)α2u(s)‖2ds+2∫t0eμ1s‖u(s)‖2β+2L2β+2ds=(μ1−2λ)∫t0eμ1s‖u(s)‖2ds+‖φ(0)‖2+2Re∫t0eμ1s(u(s),G(x,u(s−ρ)))ds+∞∑k=1∫t0eμ1s‖σ1,k+κσ2,k(u(s))‖2ds+2Re∫t0eμ1s(u(s),∞∑k=1σ1,k(x)+κ(x)σ2,k(u(s)))dWk(s). |
Taking the expectation, we have for all t≥0,
eμ1tE(‖u(t)‖2)+2E[∫t0eμ1s‖(−Δ)α2u(s)‖2ds]+2E[∫t0eμ1s‖u(s)‖pLpds]=E(‖φ(0)‖2)+(μ1−2λ)E[∫t0eμ1s‖u(s)‖2ds]+2E[∫t0eμ1sRe(u(s),G(x,u(s−ρ)))ds]+∞∑k=1E[∫t0eμ1s‖σ1,k(x)+κ(x)σ2,k(u(s))‖2ds]. | (3.7) |
For the third term on the right-hand side (3.7), by (2.1), we have
2E[∫t0eμ1sRe(u(s),G(x,u(s−ρ)))ds]≤2∫t0eμ1sE[‖u(s)‖‖G(x,u(s−ρ))‖]ds≤√2a∫t0eμ1sE(‖u(s)‖2)ds+√22a∫t0eμ1sE[‖G(x,u(s−ρ))‖2]ds≤√2a∫t0eμ1sE(‖u(s)‖2)ds+√2a∫t0eμ1s‖h(x)‖2ds+√2a∫t0eμ1sE[‖u(s−ρ)‖2]ds≤√2a(1+eμ1ρ)∫t0eμ1sE[‖u(s)‖2]ds+√2a‖h(x)‖2∫t0eμ1sds+√2aeμ1ρ∫0−ρeμ1sE[‖φ(s)‖2]ds≤√2a(1+eμ1ρ)∫t0eμ1sE[‖u(s)‖2]ds+√2eμ1taμ1‖h(x)‖2+√2aρeμ1ρE[sup−ρ≤s≤0‖φ(s)‖2]. | (3.8) |
For the forth term on the right-hand side (3.7), by (2.6), we have
∞∑k=1E[∫t0eμ1s‖σ1,k+κσ2,k(u(s))‖2ds]≤∞∑k=1E[∫t0eμ1s(2‖σ1,k‖2+2‖κσ2,k(u(s))‖2)ds]≤2μ1∞∑k=1‖σ1,k‖2eμ1t+4∞∑k=1∫t0eμ1sE[β2k‖κ‖2+γ2k‖κ‖2L∞‖u(s)‖2]ds≤2μ1∞∑k=1(‖σ1,k‖2+2β2k‖κ(x)‖2)eμ1t+4∞∑k=1γ2k‖κ(x)‖2L∞∫t0eμ1sE(‖u(s)‖2)ds. | (3.9) |
By (3.7)–(3.9), we obtain for all t≥0,
eμ1tE(‖u(t)‖2)+2E[∫t0eμ1s‖(−Δ)α2u(s)‖2ds]+2E[∫t0eμ1s‖u(s)‖2β+2L2β+2ds]≤(1+√2aρeμ1ρ)E[sup−ρ≤s≤0‖φ(s)‖2]+[μ1−2λ+√2a(1+eμ1ρ)+4∞∑k=1γ2k‖κ‖2L∞]∫t0eμ1sE[‖u‖2]ds+√2aμ1eμ1t‖h(x)‖2+2μ1∞∑k=1(‖σ1,k‖2+2β2k‖κ(x)‖2)eμ1t. | (3.10) |
By (3.2), there exists a positive constant μ1 sufficiently small such that
2μ1+√2a+√2aeμ1ρ+4∞∑k=1γ2k‖κ(x)‖2L∞≤2λ. |
Then, we have, for all t≥0,
E(‖u(t)‖2)+2∫t0eμ1(s−t)E(‖(−Δ)α2u(s)‖2)ds +μ1∫t0eμ1(s−t)E(‖u(s)‖2)ds+2∫t0eμ1(s−t)E(‖u(s)‖2β+2L2β+2)ds≤(1+√2aρeμ1ρ)E(sup−ρ≤s≤0‖φ(s)‖2)+1μ1(√2a‖h(x)‖2+2∞∑k=1(‖σ1,k‖2+2β2k‖κ(x)‖2)), |
which completes the proof of (3.4).
Integrating (3.6) on [0,t+ρ] and taking the expectation, one has
E[‖u(t+ρ)‖2]+2E[∫t+ρ0‖(−Δ)α2u(s)‖2ds]+2E[∫t+ρ0‖u(s)‖2β+2L2β+2ds]+2λE[∫t+ρ0‖u(s)‖2ds]=E[‖φ(0)‖2]+2E[∫t+ρ0Re(u(s),G(x,u(s−ρ)))ds]+∞∑k=1E[∫t+ρ0‖σ1,k+κ(x)σ2,k(u(s))ds]. | (3.11) |
For the second term on the right-hand side of (3.11), by (2.1), we have
2E[∫t+ρ0Re(u,G(x,u(s−ρ)))ds]≤2√2aE[∫t+ρ0‖u‖2ds]+√2aρE[sup−ρ≤s≤0‖φ(s)‖2]+√2(t+ρ)a‖h‖2. | (3.12) |
For the third term on the right-hand side of (3.11), one has
∞∑k=1E[∫t+ρ0‖σ1,k+κσ2,k(u(s))ds]≤2(t+ρ)∞∑k=1(‖σ1,k‖2+2β2k‖κ‖2)+4∞∑k=1γ2k‖κ(x)‖2L∞E[∫t+ρ0‖u‖2ds]. | (3.13) |
Then, by (3.2) and (3.11)–(3.13), for all t≥0, we obtain,
2E[∫t+ρ0‖(−Δ)α2u(s)‖2ds]≤(1+√2aρ)E[sup−ρ≤s≤0‖φ(s)‖2] |
+2(t+ρ)∞∑k=1(‖σ1,k‖2+2β2k‖κ(x)‖2)+√2(t+ρ)a‖h(x)‖2. |
The result then follows from (3.4).
The next lemma is used to obtain the uniform estimates of the segments of solutions in C([−ρ,0],H).
Lemma 3.2. Suppose (2.1)–(2.6) and (3.2) hold. Then, for any φ(s)∈L2(Ω,F0;C([−ρ,0],H)), the solution of (1.1) satisfies that, for all t≥ρ,
E(supt−ρ≤r≤t‖u(r)‖2)≤M2E[sup−ρ≤s≤0‖φ(s)‖2]+˜M2, |
where M2 and ˜M2 are positive constants independent of φ.
Proof. By (1.1) and integration by parts of Ito's formula and taking the real part, we get for all t≥ρ and t−ρ≤r≤t,
‖u(r)‖2+2∫rt−ρ‖(−Δ)α2u(s)‖2ds+2∫rt−ρ‖u(s)‖2β+2L2β+2ds+2λ∫rt−ρ‖u(s)‖2ds=‖u(t−ρ)‖2+2Re∫rt−ρ(u(s),G(x,u(s−ρ)))ds+∞∑k=1∫rt−ρ‖σ1,k(x)+κ(x)σ2,k(u(s)))‖2ds+2Re∞∑k=1∫rt−ρ(u(s),(σ1,k(x)+κ(x)σ2,k(u(s))dWk(s)). | (3.14) |
For the second term on the right-hand side of (3.14), by (2.1) we have, for all t≥ρ and t−ρ≤r≤t,
2Re∫rt−ρ(u(s),G(x,u(s−ρ)))ds≤2∫rt−ρ‖u(s)‖⋅‖G(x,u(s−ρ))‖ds≤∫rt−ρ‖u(s)‖2ds+∫rt−ρ‖G(x,u(s−ρ))‖2ds≤∫rt−ρ‖u(s)‖2ds+2∫rt−ρ‖h‖2ds+2a2∫rt−ρ‖u(s−ρ)‖2ds≤∫rt−ρ‖u(s)‖2ds+2ρ‖h‖2+2a2∫t−ρt−2ρ‖u(s)‖2ds. | (3.15) |
For the third term on the right-hand side of of (3.14), for all t≥ρ and t−ρ≤r≤t, by (2.6), we have
∞∑k=1∫rt−ρ‖σ1,k(x)+κ(x)σ2,k(u(s))‖2ds≤2ρ∞∑k=1‖σ1,k‖2+4ρ‖κ‖2∞∑k=1β2k+4‖κ‖2L∞∞∑k=1γ2k∫rt−ρ‖u(s)‖2ds. | (3.16) |
By (3.14)–(3.16), we obtain for all t≥ρ and t−ρ≤r≤t,
‖u(r)‖2≤c3+‖u(t−ρ)‖2+c4∫rt−2ρ‖u(s)‖2ds |
+2Re∞∑k=1∫rt−ρ(u(s),(σ1,k(x)+κ(x)σ2,k(u(s))dWk(s)), | (3.17) |
where c3=2ρ‖h‖2+2ρ∞∑k=1‖σ1,k‖2+4ρ‖κ‖2∞∑k=1β2k and c4=1+2a2+4‖κ‖2L∞∞∑k=1γ2k. By (3.17), we find that for all t≥ρ,
E[supt−ρ≤r≤t‖u(r)‖2]≤c3+E[‖u(t−ρ)‖2]+c4∫tt−2ρE[‖u(s)‖2]ds+2E[supt−ρ≤r≤t|∞∑k=1∫rt−ρ(u(s),(σ1,k(x)+κ(x)σ2,k(u(s))dWk(s))|]. | (3.18) |
For the second term and the third term on the right-hand side of (3.18), by Lemma 3.1, we deduce for all t≥ρ,
E[‖u(t−ρ)‖2]≤sups≥0E[‖u(s)‖2]≤M1E[sup−ρ≤s≤0‖φ‖2]+˜M1 | (3.19) |
and
c4∫tt−2ρE[‖u(s)‖2]ds≤2ρc4sups≥−ρE[‖u(s)‖2]≤c5E[sup−ρ≤s≤0‖φ‖2]+c5. | (3.20) |
For the last term on the right-hand side of (3.18), by Burkholder-Davis-Gundy's inequality and Lemma 3.1, we obtain for all t≥ρ,
2E[supt−ρ≤r≤t|∞∑k=1∫rt−ρ(u(s),σ1,k(x)+κ(x)σ2,k(u(s))dWk(s))|]≤2B2E[(∞∑k=1∫tt−ρ|(u(s),σ1,k+κσ2,k(u(s)))|2ds)12]≤12E[supt−ρ≤s≤t‖u(s)‖2]+2B22E[∞∑k=1∫tt−ρ‖σ1,k+κσ2,k(u(s))‖2ds]≤12E[supt−ρ≤s≤t‖u(s)‖2]+2B22(2ρ∞∑k=1‖σ1,k‖2+4ρ‖κ‖2∞∑k=1β2k)+8B22ρ‖κ‖2L∞∞∑k=1γ2ksups≥0E[‖u(s)‖2]. | (3.21) |
By Lemma 3.1 and (3.18)–(3.21), we deduce that for all t≥ρ,
E[supt−ρ≤r≤t‖u(r)‖2]≤M2E[sup−ρ≤s≤0‖φ(s)‖2]+˜M2. |
This completes the proof.
To establish the tightness of a family of distributions of solutions, we now derive uniform estimates on the tails of solutions to the problems (1.1) and (1.2).
Lemma 3.3. Suppose (2.1)–(2.6) and (3.2) hold. If φ(s)∈L2(Ω,C([−ρ,0],H)). Then, for all t≥0, the solution u of (1.1) and (1.2) satisfies
lim supm→∞supt≥−ρ∫|x|≥mE[|u(t,x)|2]dx=0. |
Proof. We suppose that θ(x):Rn→R is a smooth function with 0≤θ(x)≤1, for all x∈Rn defined by
θ(x)={0if |x|≤1,1if |x|≥2. |
For fixed m∈N, we denote that θm(x)=θ(xm). By (1.1), we have
d(θmu)+(1+iν)θm(−Δ)αudt+(1+iμ)θm|u|2βudt+λθmudt=θmG(x,u(t−ρ))dt |
+∞∑k=1θm(σ1,k+κσ2,k)dWk(t). | (3.22) |
By (3.2), We can find μ2 sufficiently small such that
μ2+2√2a+4‖κ‖2L∞∞∑k=1γ2k−2λ<0. | (3.23) |
By (3.22) and integration by parts of Ito's formula and taking the expectation, we obtain
E[‖θmu‖2]+2∫t0eμ2(s−t)E[∫Rnθ2m|u|2β+2dx]ds=e−μ2tE[‖θmφ(0)‖2]−2∫t0eμ2(s−t)E[Re(1+iν)((−Δ)α2u,(−Δ)α2(θ2mu))]ds+(μ2−2λ)∫t0eμ2(s−t)E[‖θmu‖2]ds+2∫t0eμ2(s−t)E[Re(θmu,θmG(x,u(s−ρ)))]ds+∞∑k=1∫t0eμ2(s−t)E[‖θm(σ1,k+κ(x)σ2,k(u(s)))‖2]ds. | (3.24) |
For the first term in the second row of (3.24), since φ(s)∈L2(Ω,C([−ρ,0],H)), we have for all s∈[−ρ,0], E[‖φ(0)‖2]<∞. It follows that for any ε>0, there exists a positive N1=N1(ε,φ)≥1, for all m≥N1, one has ∫|x|≥mE[φ2(0,x)]dx<ε. Consequently,
E[‖θmφ(0)‖2]=E[∫Rn|θ(xm)φ(0,x)|2dx]=E[∫|x|≥m|θ(xm)φ(0,x)|2dx]≤∫|x|≥mE[|φ(0,x)|2]dx<ε, ∀m≥N1. | (3.25) |
Now we consider the second term on the right-hand side of (3.24). We first have
−2E[Re(1+iν)((−Δ)α2u(s),(−Δ)α2(θ2mu(s)))]=−C(n,α)E[Re(1+iν)∫Rn∫Rn[u(x)−u(y)][θ2m(x)ˉu(x)−θ2m(y)ˉu(y)]|x−y|n+2α]dxdy=−C(n,α)E[Re(1+iν)∫Rn∫Rn[u(x)−u(y)][θ2m(x)(ˉu(x)−ˉu(y))+ˉu(y)(θ2m(x)−θ2m(y))]|x−y|n+2α]dxdy=−C(n,α)E[Re(1+iν)∫Rn∫Rnθ2m(x)|u(x)−u(y)|2|x−y|n+2αdxdy]−C(n,α)E[Re(1+iν)∫Rn∫Rn(u(x)−u(y))(θ2m(x)−θ2m(y))ˉu(y)|x−y|n+2αdxdy]≤−C(n,α)E[Re(1+iν)∫Rn∫Rn(u(x)−u(y))(θ2m(x)−θ2m(y))ˉu(y)|x−y|n+2αdxdy]≤C(n,α)√1+ν2E[|∫Rn∫Rn(u(x)−u(y))(θ2m(x)−θ2m(y))ˉu(y)|x−y|n+2αdxdy|]≤2C(n,α)√1+ν2E[∫Rn|ˉu(y)|(∫Rn|(u(x)−u(y))(θm(x)−θm(y))||x−y|n+2αdx)dy]≤2C(n,α)√1+ν2E[‖u(s)‖(∫Rn(∫Rn|(u(x)−u(y))(θm(x)−θm(y))||x−y|n+2αdx)2dy)12]≤2C(n,α)√1+ν2E[‖u(s)‖(∫Rn(∫Rn|u(x)−u(y)|2|x−y|n+2αdx∫Rn|(θm(x)−θm(y))|2|x−y|n+2αdx)dy)12]. | (3.26) |
We now prove the following inequality:
∫Rn|(θm(x)−θm(y))|2|x−y|n+2αdx≤c6m2α. | (3.27) |
Let x−y=h and hm=z, then, we obtain,
∫Rn|(θm(x)−θm(y))|2|x−y|n+2αdx=∫Rn|θ(y+hm)−θ(ym)|2|h|n+2αdh=∫Rn|θ(ym+z)−θ(ym)|2mn+2α|z|n+2αmndz=1m2α∫Rn|θ(ym+z)−θ(ym)|2|z|n+2αdz=1m2α∫|z|≤1|θ(ym+z)−θ(ym)|2|z|n+2αdz+1m2α∫|z|>1|θ(ym+z)−θ(ym)|2|z|n+2αdz≤c∗6m2α∫|z|≤1|z|2|z|n+2αdz+4m2α∫|z|>11|z|n+2αdz≤c∗6m2α∫|z|≤11|z|n+2α−2dz+4m2α∫|z|>11|z|n+2αdz≤c∗6ˉc6m2α+4˜c6m2α=c∗6ˉc6+4˜c6m2α. | (3.28) |
This proves (3.27) with c6:=c∗6ˉc6+4˜c6. By (3.26) and (3.27), we obtain,
−2E[Re(1+iν)((−Δ)α2u(s),(−Δ)α2θ2mu(s))]≤2√c6(1+ν2)C(n,α)m−αE[‖u(s)‖√∫Rn∫Rn|u(x)−u(y)|2|x−y|n+2αdxdy]≤√c6(1+ν2)C(n,α)m−α(E(‖u(s)‖2)+E(∫Rn∫Rn|u(x)−u(y)|2|x−y|n+2αdxdy))≤√c6(1+ν2)C(n,α)m−αE(‖u(s)‖2)+2√c6(1+ν2)m−αE(‖(−Δ)α2u(s)‖2). | (3.29) |
By (3.29), for the second term on the right-hand side of (3.24), we get
−2∫t0eμ2sE[Re(1+iν)((−Δ)α2u(s),(−Δ)α2θ2mu(s))]ds≤√c6(1+ν2)C(n,α)m−α∫t0eμ2sE[‖u(s)‖2]ds+2√c6(1+ν2)m−α∫t0eμ2sE[‖(−Δ)α2u(s)‖2]ds. | (3.30) |
By Lemma 3.1, we have
√c6(1+ν2)C(n,α)m−α∫t0eμ2(s−t)E[‖u(s)‖2]ds≤√c6(1+ν2)C(n,α)m−α[M1E[sup−ρ≤s≤0‖φ(s)‖2]+˜M1]∫t0eμ2(s−t)ds≤√c6(1+ν2)C(n,α)m−α1μ2[M1E[sup−ρ≤s≤0‖φ(s)‖2]+˜M1]. | (3.31) |
By (3.31), we deduce that there exists N2(ε,φ)≥N1, for all t≥0 and m≥N2,
√c6(1+ν2)C(n,α)m−α∫t0eμ2(s−t)E[‖u(s)‖2]ds<ε. |
By Lemma 3.1, there exists N3(ε,φ)≥N2 such that for all t≥0 and m≥N3,
2√c6(1+ν2)m−α∫t0eμ2(s−t)E[‖(−Δ)α2u‖2]ds≤2√c6(1+ν2)m−α[M1E[sup−ρ≤s≤0‖φ(s)‖2]+˜M1]<ε. |
For the forth term on the right-hand side of (3.24), we obtain that there exists N4(ε,φ)≥N3, for all t≥0 and m≥N4,
2∫t0eμ2(s−t)E[Re(θmu,θmG(x,u(s−ρ)))]ds≤√2a∫t0eμ2(s−t)E[‖θmu(s)‖2]ds+1√2a∫t0eμ2(s−t)E[‖θmG(x,u(s−ρ))‖2]ds≤√2aμ2∫|x|≥mh2(x)dx+√2a∫0−ρeμ2(s−t)E[‖θmφ(s)‖2]ds+2√2a∫t0eμ2(s−t)E[‖θmu(s)‖2]ds≤√2aμ2ε+√2a∫0−ρeμ2(s−t)E[‖θmφ(s)‖2]ds+2√2a∫t0eμ2(s−t)E[‖θmu(s)‖2]ds. |
Since {φ(s)∈L2(Ω,H)|s∈[−ρ,0]} is compact, it has a open cover of balls with radius √ε2 which denoted by {B(φi,√ε2)}li=1. Since φi=φ(si)∈L2(Ω;C([−ρ,0],H)) for i=1,2,⋯,l, we obtain that for given ε>0,
{φ(s)∈L2(Ω;C([−ρ,0],H))}⊆∪li=1{X∈L2(Ω,H)|‖X−φi‖L2(Ω,H)<√ε2}. |
Since φi∈L2(Ω,H), there exists a positive constant N5=N5(ε,φ)≥N4, for m≥N5, we have
supi=1,2,⋯,l∫|x|≥mE[|φ(si,x)|2]dx<ε4. |
Then,
sups∈[−ρ,0]∫|x|≥mE[|φ(s,x)|2]dx<ε2,∀m≥N5. |
Consequently, one has
2∫t0eμ2(s−t)E[Re(θmu,θmG(x,u(s−ρ)))]ds≤√2aμ2ε+√2aρε2+2√2a∫t0eμ2(s−t)E[‖θmu(s)‖2]ds. | (3.32) |
For the fifth term on the right-hand side of (3.24), by (2.6), we obtain
∞∑k=1∫t0eμ2(s−t)E[‖θm(σ1,k+κ(x)σ2,k(u(s)))‖2]ds≤2∞∑k=1∫t0eμ2(s−t)‖θmσ1,k‖2ds+2∞∑k=1∫t0eμ2(s−t)E[‖θmκ(x)σ2,k(u(s))‖2]ds≤2μ2∞∑k=1∫|x|≥m|σ1,k(x)|2dx+4μ2∞∑k=1β2k∫|x|≥mκ2(x)dx+4‖κ(x)‖2L∞∞∑k=1γ2k∫t0eμ2(s−t)E[‖θmu(s)‖2]ds. |
Since ∞∑k=1‖σ1,k‖2<∞ and κ(x)∈L2(Rn)⋂L∞(Rn), there exists N6=N6(ε,φ)≥N5, for all t≥0 and m≥N6, we have
∞∑k=1∫|x|≥m|σ1,k(x)|2dx+∫|x|≥mκ2(x)dx<ε. |
Consequently, for the fifth term on the right-hand side of (3.24), we get for all t≥0 and m≥N6,
∞∑k=1∫t0eμ2(s−t)E[‖θm(σ1,k+κσ2,k)‖2]ds≤2μ2(1+2∞∑k=1β2k)ε |
+4‖κ‖2L∞∑∞k=1γ2k∫t0eμ2(s−t)E[‖θmu(s)‖2]ds. |
Therefore, for all t≥0 and m≥N6,
E[‖θmu(t)‖2]≤[2+e−μ2t+√2aμ2+√22aρ+2μ2(1+2∞∑k=1β2k)]ε |
+(μ2−2λ+2√2a+4‖κ‖2L∞∑∞k=1γ2k)∫t0eμ2(s−t)E[‖θmu(s)‖2]ds. |
Taking the limit in the above equation and by (3.23), we have
lim supm→∞supt≥−ρ∫|x|≥mE[|u(t,x)|2]dx=0, |
which completes the proof.
Lemma 3.4. Suppose (2.1)–(2.6) and (3.2) hold. If φ(s)∈L2(Ω,C([−ρ,0],H)), then the solution u of (1.1) and (1.2) satisfies
lim supm→∞supt≥0E[supr∈[t−ρ,t]∫|x|≥m|u(r,x)|2dx]=0. |
Proof. By (3.22) and integration by parts of Ito's formula and taking the real part, for all t≥ρ and r∈[t−ρ,t], we have
eμ2r‖θmu(r)‖2+2∫rt−ρeμ2s∫Rnθ2m|u|2β+2dxds=eμ2(t−ρ)‖θmu(t−ρ)‖2−2∫rt−ρeμ2sRe(1+iν)((−Δ)α2u(s),(−Δ)α2θ2mu(s))ds+(μ2−2λ)∫rt−ρeμ2s‖θmu(s)‖2ds+2Re∫rt−ρeμ2s(θmu(s),θmG(x,u(s−ρ)))ds+∞∑k=1∫rt−ρeμ2s‖θm(σ1,k+κ(x)σ2,k(u(s)))‖2ds+2Re∞∑k=1∫rt−ρeμ2s(θmu(s),θm(σ1,k+κσ2,k(u(s))))dWk(s). | (3.33) |
By (3.33), we deduce,
E[supt−ρ≤r≤t‖θmu(r)‖2]≤E[‖θmu(t−ρ)‖2]−2E[supt−ρ≤r≤t∫rt−ρeμ2(s−r)Re(1+iν)((−Δ)α2u,(−Δ)α2θ2mu)ds]+|μ2−2λ|E[supt−ρ≤r≤t∫rt−ρ‖θmu‖2eμ2(s−r)ds]+2E[supt−ρ≤r≤t∫rt−ρeμ2(s−r)‖θmu‖⋅‖θmG(x,u(s−ρ))‖ds]+∞∑k=1E[supt−ρ≤r≤t∫rt−ρeμ2(s−r)‖θm(σ1,k+κσ2,k(u(s)))‖2ds]+2E[supt−ρ≤r≤t|∞∑k=1∫rt−ρeμ2(s−r)(θmu(s),θm(σ1,k+κσ2,k(u(s))))dWk(s)|]. | (3.34) |
For the first term on the right-hand side of (3.34), by Lemma 3.3, one has for any ε>0, there exists ˜N1(ε,φ)≥1 such that for all m≥˜N1 and t≥ρ,
E[‖θmu(t−ρ)‖2]≤∫|x|≥mE[|u(t−ρ,x)|2]dx<ε. | (3.35) |
For the second term on the right-hand side of (3.34), by (3.29), we have
−2E[supt−ρ≤r≤t∫rt−ρeμ2(s−r)Re(1+iν)((−Δ)α2u(s),(−Δ)α2θ2mu(s))ds]≤2√c6(1+ν2)C(n,α)m−αE[supt−ρ≤r≤t(∫rt−ρeμ2(s−r)‖u(s)‖‖(−Δ)α2u(s)‖ds)]≤2√c6(1+ν2)C(n,α)m−αeμ2ρE[(∫tt−ρeμ2(s−t)‖u(s)‖‖(−Δ)α2u(s)‖ds)]≤√c6(1+ν2)C(n,α)m−αeμ2ρ{∫tt−ρeμ2(s−t)E[‖u‖2]ds+E[∫tt−ρeμ2(s−t)‖(−Δ)α2u‖2ds]}≤√c6(1+ν2)C(n,α)m−αeμ2ρ{ρsups∈[t−ρ,t]E[‖u(s)‖2]+E[∫tt−ρeμ2(s−t)‖(−Δ)α2u‖2ds]}. | (3.36) |
By Lemma 3.1 and (3.36), we deduce that there exists ˜N2(ε,φ)≥˜N1 such that for all m≥˜N2 and t≥ρ,
−2E[supt−ρ≤r≤t∫rt−ρeμ2(s−r)Re(1+iν)((−Δ)α2u(s),(−Δ)α2θ2mu(s))ds]<ε. | (3.37) |
For the third term on the right-hand side of (3.34), by Lemma 3.3, we obtain that for all m≥˜N2 and t≥ρ,
|μ2−2λ|E[supt−ρ≤r≤t∫rt−ρ‖θmu(s)‖2eμ2(s−r)ds]≤|μ2−2λ|E[∫tt−ρ‖θmu(s)‖2ds] |
≤|μ2−2λ|ρsupt−ρ≤s≤tE[‖θmu(s)‖2]<|μ2−2λ|ρε. | (3.38) |
For the forth term on the right-hand side of (3.34), by (2.1), we obtain
2E[supt−ρ≤r≤t∫rt−ρeμ2(s−r)‖θmu(s)‖⋅‖θmG(x,u(s−ρ))‖ds]≤∫tt−ρE[‖θmu(s)‖2]ds+2ρ‖θmh‖2+2a2∫t−ρt−2ρE[‖θmu(s)‖2]ds≤ρsupt−ρ≤s≤tE[‖θmu(s)‖2]+2ρ‖θmh‖2+2a2ρsupt−2ρ≤s≤t−ρE[‖θmu(s)‖2], |
which along with Lemma 3.3, we deduce that there exists ˜N3(ε,φ)≥˜N2 such that for all m≥˜N3 and t≥ρ,
2E[supt−ρ≤r≤t∫rt−ρeμ2(s−r)‖θmu(s)‖⋅‖θmG(x,u(s−ρ))‖ds]<(3+2a2)ρε. | (3.39) |
For the fifth term on the right-hand side of (3.34), by (2.6), we have
∞∑k=1E[supt−ρ≤r≤t∫rt−ρeμ2(s−r)‖θm(σ1,k+κσ2,k(u(s)))‖2ds]≤2ρ∞∑k=1‖θmσ1,k‖2+2ρ∞∑k=1supt−ρ≤s≤tE[‖θmκ(x)σ2,k(u(s))‖2]≤2ρ∞∑k=1∫|x|≥m|σ1,k(x)|2dx+4ρ∞∑k=1β2k∫|x|≥m|κ(x)|2dx+4ρ‖κ(x)‖2L∞∞∑k=1γ2ksupt−ρ≤s≤tE[‖θmu(s)‖2]. |
By the condition κ(x)∈L2(Rn)⋂L∞(Rn), (2.4) and Lemma 3.3, we deduce that there exists ˜N4(ε,φ)≥˜N3 such that for all m≥˜N4 and t≥ρ,
∞∑k=1E[supt−ρ≤r≤t∫rt−ρeμ2(s−r)‖θm(σ1,k+κσ2,k(u(s)))‖2ds]<2ρ(1+λ+2∞∑k=1β2k)ε. | (3.40) |
For the sixth term on the right-hand side of (3.34), by (2.6), (3.40) and Burkholder-Davis-Gundy's inequality, we have,
2E[supt−ρ≤r≤t|∞∑k=1∫rt−ρeμ2(s−r)(θmu(s),θm(σ1,k+κσ2,k(u(s))))dWk(s)|]≤2e−μ2(t−ρ)E[supt−ρ≤r≤t|∞∑k=1∫rt−ρeμ2s(θmu(s),θmσ1,k+θmκ(x)σ2,k(u(s)))dWk(s)|]≤2˜B2e−μ2(t−ρ)E[(∫tt−ρe2μ2s∞∑k=1|(θmu(s),θmσ1,k+θmκ(x)σ2,k(u(s)))|2ds)12]≤2˜B2e−μ2(t−ρ)E[supt−ρ≤s≤t‖θmu(s)‖(∫tt−ρe2μ2s∞∑k=1‖θmσ1,k+θmκσ2,k(u(s))‖2ds)12]≤12E[supt−ρ≤s≤t‖θmu(s)‖2]+2˜B22E[e2μ2ρ∫tt−ρe2μ2(s−t)∞∑k=1‖θmσ1,k+θmκ(x)σ2,k(u(s))‖2ds]≤12E[supt−ρ≤s≤t‖θmu(s)‖2]+2˜B22e2μ2ρ∞∑k=1E[supt−ρ≤r≤t∫rt−ρeμ2(s−r)‖θmσ1,k+θmκ(x)σ2,k(u(s))‖2ds]≤12E[supt−ρ≤s≤t‖θmu(s)‖2]+4ρ(1+λ+2∞∑k=1β2k)˜B22e2μ2ρε. |
Above all, for all m≥˜N4 and t≥ρ, we obtain,
E[supt−ρ≤r≤t‖θmu(r)‖2]≤[4+2|μ2−2λ|ρ+(6+4a2)ρ+4ρ(1+2˜B22e2μ2ρ)(1+λ+2∞∑k=1β2k)]ε. |
Therefore, we conclude
lim supm→∞supt≥0E[supt−ρ≤r≤t∫|x|≥m|u(r,x)|2dx]=0. |
Lemma 3.5. Suppose (2.1)–(2.6) and (3.1) hold. If φ(s)∈L2(Ω,C([−ρ,0],H)), then there exists a positive constant μ3 such that the solution u of (1.1) and (1.2) satisfies
supt≥−ρE[‖u(t)‖2p]+supt≥0E[∫t0eμ3(s−t)‖u(s)‖2p−2‖(−Δ)α2u(s)‖2ds] ≤(1+aρeμρ2p(4p−2)2p−12p)E[‖φ‖2pCH]+M3, | (3.41) |
where M3 is a positive constant independent of φ.
Proof. By (3.1), there exist positive constants μ and ϵ1 such that
μ+aeμρ2p21−12p(2p−1)2p−12p+4(p−1)(2p−1)ϵ2p2p−21∞∑k=1(‖σ1,k‖2+‖κ‖2β2k) +4θ(2p−1)‖κ‖2L∞∞∑k=1γ2k≤2pλ. | (3.42) |
Given n∈N, let τn be a stopping time as defined by
τn=inf{t≥0:‖u(t)‖>n}, |
and as usual, we set τn=+∞ if {t≥0:‖u(t)‖>n}=∅. By the continuity of solutions, we have
limn→∞τn=+∞. |
Applying Ito's formula, we obtain
d(‖u(t)‖2p)=d((‖u(t)‖2)p)=p‖u(t)‖2(p−1)d(‖u(t)‖2)+2p(p−1)‖u(t)‖2(p−2) ×∞∑k=1|(u(t),σ1,k+κσ2,k(u(t)))|2dt. | (3.43) |
Substituting (3.6) into (3.43), we infer
d(‖u(t)‖2p)=−2p‖u(t)‖2(p−1)‖(−Δ)α2u(t)‖2dt−2p‖u(t)‖2(p−1)‖u(t)‖2β+2L2β+2dt−2pλ‖u(t)‖2pdt +2p‖u(t)‖2(p−1)Re(u(t),G(x,u(t−ρ)))dt +p‖u(t)‖2(p−1)∞∑k=1‖σ1,k(x)+κ(x)σ2,k(u(t))‖2dt +2p‖u(t)‖2(p−1)Re(u(t),∞∑k=1σ1,k(x)+κ(x)σ2,k(u(t)))dWk(t) +2p(p−1)‖u(t)‖2(p−2)∞∑k=1|(u(t),σ1,k+κσ2,k(u(t)))|2dt. | (3.44) |
We also get the formula
d(eμt‖u(t)‖2p)=μeμt‖u(t)‖2pdt+eμtd(‖u(t)‖2p). | (3.45) |
Substituting (3.44) into (3.45) and integrating on (0,t∧τn) with t≥0, we deduce
eμ(t∧τn)‖u(t∧τn)‖2p+2p∫t∧τn0eμs‖u(s)‖2(p−1)‖(−Δ)α2u(s)‖2ds=−2p∫t∧τn0eμs‖u(s)‖2(p−1)‖u(s)‖2β+2L2β+2ds+‖φ(0)‖2p+(μ−2pλ)∫t∧τn0eμs‖u(s)‖2pds +2p∫t∧τn0eμs‖u(s)‖2(p−1)Re(u(s),G(x,u(s−ρ)))ds +p∞∑k=1∫t∧τn0eμs‖u(s)‖2(p−1)‖σ1,k+κσ2,k(u(s))‖2ds +2p∞∑k=1∫t∧τn0eμs‖u(t)‖2(p−1)Re(u(s),σ1,k+κσ2,k(u(s)))dWk(s) +2p(p−1)∞∑k=1∫t∧τn0eμs‖u(s)‖2(p−2)|(u(s),σ1,k+κσ2,k(u(s)))|2ds. | (3.46) |
Taking the expectation, we obtain for t≥0,
E[eμ(t∧τn)‖u(t∧τn)‖2p]+2pE[∫t∧τn0eμs‖u(s)‖2(p−1)‖(−Δ)α2u(s)‖2ds]=−2pE[∫t∧τn0eμs‖u(s)‖2(p−1)‖u(s)‖2β+2L2β+2ds]+E[‖φ(0)‖2p]+(μ−2pλ)E[∫t∧τn0eμs‖u(s)‖2pds] +2pE[∫t∧τn0eμs‖u(s)‖2(p−1)Re(u(s),G(x,u(s−ρ)))ds] +p∞∑k=1E[∫t∧τn0eμs‖u(s)‖2(p−1)‖σ1,k+κσ2,k(u(s))‖2ds] +2p(p−1)∞∑k=1E[∫t∧τn0eμs‖u(s)‖2(p−2)|(u(s),σ1,k+κσ2,k(u(s)))|2ds]≤E[‖φ(0)‖2p]+(μ−2pλ)E[∫t∧τn0eμs‖u(s)‖2pds] +2pE[∫t∧τn0eμs‖u(s)‖2(p−1)Re(u(s),G(x,u(s−ρ)))ds] +p∞∑k=1E[∫t∧τn0eμs‖u(s)‖2(p−1)‖σ1,k+κσ2,k(u(s))‖2ds] +2p(p−1)∞∑k=1E[∫t∧τn0eμs‖u(s)‖2(p−2)|(u(s),σ1,k+κσ2,k(u(s)))|2ds]. | (3.47) |
Next, we estimate the terms on the right-hand side of (3.47).
For the third term on the right-hand side of (3.47), by Young's inequality and (2.1), we infer
2θE[∫t∧τn0eμs‖u(s)‖2(p−1)Re(u(s),G(x,u(s−ρ)))ds]≤2θE[∫t∧τn0eμs‖u(s)‖2p−1‖G(x,u(s−ρ))‖2ds]≤aeμρ2p21−12p(2p−1)2p−12pE[∫t∧τn0eμs‖u(s)‖2pds] +(2p−122p−1a2peμρ)2p−12pE[∫t∧τn0eμs‖G(x,u(s−ρ))‖2ds]≤aeμρ2p21−12p(2p−1)2p−12pE[∫t∧τn0eμs‖u(s)‖2pds] +22p−1(2p−122p−1a2peμρ)2p−12pE[∫t∧τn0eμs(‖h‖2p+a2p‖u(s−ρ)‖2p)ds]≤aeμρ2p21−12p(2p−1)2p−12pE[∫t∧τn0eμs‖u(s)‖2pds] +1μ(4p−2a2peμρ)2p−12p‖h‖2peμt+aρeμρ2p(4p−2)2p−12pE[‖φ‖2pCH]. | (3.48) |
For the forth term on the right-hand side of (3.47), we infer
p∞∑k=1E[∫t∧τn0eμs‖u(s)‖2(p−1)‖σ1,k+κσ2,k(u(s))‖2ds]≤2p∞∑k=1E[∫t∧τn0eμs‖u(s)‖2(p−1)‖σ1,k‖2ds] +2p∞∑k=1E[∫t∧τn0eμs‖u(s)‖2(p−1)‖κσ2,k(u(s))‖2ds]. | (3.49) |
For the first term on the right-hand side of (3.49), we have
2p∞∑k=1E[∫t∧τn0eμs‖u(s)‖2(p−1)‖σ1,k‖2ds]≤2(p−1)ϵ2p2p−21∞∑k=1‖σ1,k‖2E[∫t∧τn0eμs‖u(s)‖2pds]+2μϵp1∞∑k=1‖σ1,k‖2eμt. | (3.50) |
For the second term on the right-hand side of (3.49), we have
2p∞∑k=1E[∫t∧τn0eμs‖u(s)‖2(p−1)‖κσ2,k(u(s))‖2ds]≤4p‖κ‖2∞∑k=1β2kE[∫t∧τn0eμs‖u(s)‖2(p−1)ds]+4p‖κ‖2L∞∞∑k=1γ2kE[∫t∧τn0eμs‖u(s)‖2pds]≤4(p−1)ϵ2p2p−21‖κ‖2∞∑k=1β2kE[∫t∧τn0eμs‖u(s)‖2pds] +4μϵp1‖κ‖2∞∑k=1β2keμt+4p‖κ‖2L∞∞∑k=1γ2kE[∫t∧τn0eμs‖u(s)‖2pds]. | (3.51) |
By (3.49)–(3.51), we obtain
p∞∑k=1E[∫t∧τn0eμs‖u(s)‖2(p−1)‖σ1,k+κσ2,k(u(s))‖2ds]≤[4(p−1)ϵ2p2p−21∞∑k=1(‖σ1,k‖2+‖κ‖2β2k)+4p‖κ‖2L∞∞∑k=1γ2k]E[∫t∧τn0eμs‖u(s)‖2pds] +2μϵp1∞∑k=1(‖σ1,k‖2+2‖κ‖2β2k))eμt. | (3.52) |
For the fifth term on the right-hand side of (3.47), applying (3.52), we have
2p(p−1)∞∑k=1E[∫t∧τn0eμs‖u(s)‖2(p−2)|(u(s),σ1,k+κσ2,k(u(s)))|2ds]≤2p(p−1)∞∑k=1E[∫t∧τn0eμs‖u(s)‖2p−2‖σ1,k+κσ2,k(u(s))‖2ds]≤[8(p−1)2ϵ2p2p−21∞∑k=1(‖σ1,k‖2+‖κ‖2β2k)+8p(p−1)‖κ‖2L∞∞∑k=1γ2k]E[∫t∧τn0eμs‖u(s)‖2pads] +4(p−1)μϵp1∞∑k=1(‖σ1,k‖2+2‖κ‖2β2k))eμt. | (3.53) |
From (3.47), (3.48), (3.52) and (3.53), we obtain that for t\geq 0 ,
\begin{align} \begin{split} &\ \ \ \ \mathbb{E}\left[e^{\mu(t\wedge\tau_n)}\|u(t\wedge\tau_n)\|^{2p}\right]+2p\mathbb{E}\left[\int_0^{t\wedge\tau_n}e^{\mu s}\|u(s)\|^{2(p-1)}\|(-\Delta)^{\frac{\alpha}{2}}u(s)\|^{2}ds\right]\\& \leq\left(1+a\rho e^{\frac{\mu\rho}{2p}}(4p-2)^{\frac{2p-1}{2p}} \right)\mathbb{E}\left[\|\varphi\|^{2p}_{C_H}\right]\\&\ \ \ +\left(\mu-2p\lambda+ae^{\frac{\mu\rho}{2p}}2^{1-\frac{1}{2p}}(2p-1)^{\frac{2p-1}{2p}} +4(p-1)(2p-1)\epsilon_1^{\frac{2p}{2p-2}}\right.\\&\ \ \ \ \left.\times\sum\limits^\infty_{k = 1}(\|\sigma_{1,k}\|^2 +\|\kappa\|^2\beta^2_k)+4p(2p-1)\|\kappa\|^2_{L^\infty}\sum\limits^\infty_{k = 1}\gamma^2_k\right)\mathbb{E}\left[\int_0^{t\wedge\tau_n}e^{\mu s}\|u(s)\|^{2p}ds\right]\\&\ \ \ \ \ \ + \frac 1\mu\left(\frac{4p-2}{a^{2p}e^{\mu\rho}}\right)^{\frac{2p-1}{2p}}\|h\|^{2p}e^{\mu t}+\frac{4(p-1)}{\mu\epsilon_1^p}\sum\limits^\infty_{k = 1}(\|\sigma_{1,k}\|^2 +2\|\kappa\|^2\beta^2_k))e^{\mu t}. \end{split} \end{align} | (3.54) |
Then by (3.42) and (3.54), we obtain that for t\geq 0 ,
\begin{align} \begin{split} &\ \ \ \ \mathbb{E}\left[e^{\mu(t\wedge\tau_n)}\|u(t\wedge\tau_n)\|^{2p}\right]+2p\mathbb{E}\left[\int_0^{t\wedge\tau_n}e^{\mu s}\|u(s)\|^{2(p-1)}\|(-\Delta)^{\frac{\alpha}{2}}u(s)\|^{2}ds\right]\\& \leq\left(1+a\rho e^{\frac{\mu\rho}{2p}}(4p-2)^{\frac{2p-1}{2p}} \right)\mathbb{E}\left[\|\varphi\|^{2p}_{C_H}\right]+ \frac 1\mu\left(\frac{4p-2}{a^{2pa}e^{\mu\rho}}\right)^{\frac{2p-1}{2p}}\|h\|^{2p}e^{\mu t}\\&\ \ \ +\frac{4(p-1)}{\mu\epsilon_1^p}\sum\limits^\infty_{k = 1}(\|\sigma_{1,k}\|^2 +2\|\kappa\|^2\beta^2_k))e^{\mu t}. \end{split} \end{align} | (3.55) |
Letting n\rightarrow \infty , by Fatou's Lemma, we deduce that for t\geq 0 ,
\begin{align*} &\ \ \ \ \mathbb{E}\left[e^{\mu t}\|u(t)\|^{2p}\right]+2p\mathbb{E}\left[\int_0^{t}e^{\mu s}\|u(s)\|^{2(p-1)}\|(-\Delta)^{\frac{\alpha}{2}}u(s)\|^{2}ds\right]\\& \leq\left(1+a\rho e^{\frac{\mu\rho}{2p}}(4\theta-2)^{\frac{2\theta-1}{2p}} \right)\mathbb{E}\left[\|\varphi\|^{2p}_{C_H}\right]+ \frac 1\mu\left(\frac{4p-2}{a^{2p}e^{\mu\rho}}\right)^{\frac{2p-1}{2p}}\|h\|^{2p}e^{\mu t}\\&\ \ \ +\frac{4(p-1)}{\mu\epsilon_1^p}\sum\limits^\infty_{k = 1}(\|\sigma_{1,k}\|^2 +2\|\kappa\|^2\beta^2_k))e^{\mu t}. \end{align*} |
Hence, we have for t\geq 0 ,
\begin{align*} &\ \ \ \ \mathbb{E}\left[\|u(t)\|^{2p}\right]+2p\mathbb{E}\left[\int_0^{t}e^{\mu (s-t)}\|u(s)\|^{2(p-1)}\|(-\Delta)^{\frac{\alpha}{2}}u(s)\|^{2}ds\right]\\& \leq\left(1+a\rho e^{\frac{\mu\rho}{2p}}(4p-2)^{\frac{2p-1}{2p}} \right)\mathbb{E}\left[\|\varphi\|^{2p}_{C_H}\right]+ \frac 1\mu\left(\frac{4p-2}{a^{2p}e^{\mu\rho}}\right)^{\frac{2p-1}{2p}}\|h\|^{2p}\\&\ \ \ +\frac{4(p-1)}{\mu\epsilon_1^p}\sum\limits^\infty_{k = 1}(\|\sigma_{1,k}\|^2 +2\|\kappa\|^2\beta^2_k)). \end{align*} |
This implies the desired estimate.
In this section, we establish the uniform estimates of solutions of problems (1.1) and (1.2) with initial data in C([-\rho, 0], V) . To the end, we assume that for each k\in\mathbb{N} , the function \sigma_{1, k}\in V and
\begin{align} \sum\limits^\infty_{k = 1}\|\sigma_{1,k}\|^2_V < \infty. \end{align} | (4.1) |
Furthermore, we assume that the function \kappa\in V and there exists a constant C > 0 such that
\begin{align} |\nabla\kappa(x)|\leq C. \end{align} | (4.2) |
In the sequel, we further assume that the constant a , \hat{\gamma}_k in (2.7) are sufficiently small in the sense that there exists a constant p\geq 2 such that
\begin{align} \hat{a}2^{1-\frac{1}{2p}}(2p-1)^{\frac{2p-1}{2p}}+2p(2p-1)\|\kappa\|^2_{L^\infty}\sum\limits^\infty_{k = 1}(\beta^2_k+\hat{\beta}^2_k+\gamma_k^2+\hat{\gamma}^2_k) < p\frac\lambda 2. \end{align} | (4.3) |
By (4.3), we can find
\begin{align} \sqrt{2}\hat{a}+2\|\kappa\|^2_{L^\infty}\sum\limits^\infty_{k = 1}\hat{\gamma}^2_k < \frac \lambda 2. \end{align} | (4.4) |
Lemma 4.1. Suppose (2.1)–(2.7) and (4.4) hold. If \varphi(s)\in L^2(\Omega; C([-\rho, 0], V)) , then, for all t\geq0 , there exists a positive constant \mu_4 such that the solution u of (1.1) and (1.2) satisfies
\begin{align} \begin{split} \sup\limits_{s\geq -\rho}\mathbb{E}[\|\nabla u(t)\|^2]+\sup\limits_{s\geq 0}\mathbb{E}\left[\int^t_0e^{\mu_4(s-t)}\|(-\Delta)^{\frac{\alpha+1}{2}}u(s)\|^2ds\right] \leq M_4\left(\mathbb{E}[\|\varphi\|^2_{C_V}]+1\right), \end{split} \end{align} | (4.5) |
where M_4 is a positive constant independent of \varphi .
Proof. By (4.4), there exists a positive constant \mu_1 such that
\begin{align} \mu_1-2\lambda+8\|\kappa\|^2_{L^\infty}\sum\limits^\infty_{k = 1}\hat{\gamma}^2_k < 0. \end{align} | (4.6) |
By (1.1) and applying Ito's formula to e^{\mu_1 t}\|\nabla u(t)\|^2 , we have for t\geq 0 ,
\begin{align*} &\ \ \ \ e^{\mu_1t}\|\nabla u(t)\|^2+2\int^t_0e^{\mu_1s}\|(-\Delta)^{\frac{\alpha+1}{2}}u(s)\|^2ds+2\int^t_0e^{\mu_1s}\mbox{Re}\left((1+\text{i}\mu)|u(s)|^{2\beta}u(s),-\Delta u(s)\right)ds\\& = (\mu_1-2\lambda)\int^t_0e^{\mu_1s}\|\nabla u(s)\|^2ds+\|\nabla\varphi(0)\|^2+2\mbox{Re}\int^t_0e^{\mu_1s}(G(x,u(s-\rho)),-\Delta u(s))ds\\& \ \ \ \ +\sum\limits^\infty_{k = 1}\int^t_0e^{\mu_1s}\|\nabla(\sigma_{1,k}+\kappa\sigma_{2,k}(u(s)))\|^2ds\\& \ \ \ \ \ +2\sum\limits^\infty_{k = 1}\mbox{Re}\int^t_0e^{\mu_1s}\left(\sigma_{1,k}(x)+\kappa(x)\sigma_{2,k}(u(s)),-\Delta u(s)\right)dW_k(s). \end{align*} |
Taking the expectation, we have for all t\geq0 ,
\begin{align} \begin{split} &\ \ \ \ e^{\mu_1t}\mathbb{E}[\|\nabla u(t)\|^2]\!\!+\!\!2\mathbb{E}\left[\int^t_0e^{\mu_1s}\|(\!\!-\!\!\Delta)^{\frac{\alpha+1}{2}}u(s)\|^2ds\right] \!\!+\!\!2\mathbb{E}\left[\int^t_0e^{\mu_1s}\mbox{Re}\left((1\!\!+\!\!\text{i}\mu)|u(s)|^{2\beta}u(s),\!\!-\!\!\Delta u(s)\right)ds\right]\\& = (\mu_1-2\lambda)\mathbb{E}\left[\int^t_0e^{\mu_1s}\|\nabla u(s)\|^2ds\right]\!\!+\!\!\mathbb{E}\left[\|\nabla\varphi(0)\|^2\right]\!\!+\!\!2\mathbb{E}\left[\mbox{Re}\int^t_0e^{\mu_1s}(G(x,u(s-\rho)),-\Delta u(s))ds\right]\\& \ \ \ \ \!\!+\!\!\sum\limits^\infty_{k = 1}\mathbb{E}\left[\int^t_0e^{\mu_1s}\|\nabla(\sigma_{1,k}\!\!+\!\!\kappa\sigma_{2,k}(u(s)))\|^2ds\right]. \end{split} \end{align} | (4.7) |
First, we estimate the third term on the left-hand side of (4.7). Applying integrating by parts, we have
\begin{align} \begin{split} &\ \ \ \ \mbox{Re}\left((1+\text{i}\mu)|u|^{2\beta}u,\Delta u\right)\\& = -\mbox{Re}(1+\text{i}\mu)\int_{\mathbb{R}^n}\left((\beta+1)|u|^{2\beta}|\nabla u|^2+\beta|u|^{2(\beta-1)}(u\nabla \overline{u})^2\right)dx\\& = \int_{\mathbb{R}^n}|u|^{2(\beta-1)}\left(-(\beta+1)|u|^{2}|\nabla u|^2+\frac{\beta(1+\text{i}\mu)}{2}(u\nabla \overline{u})^2+\frac{\beta(1-\text{i}\mu)}{2}(\overline{u}\nabla u)^2\right)dx\\& = \int_{\mathbb{R}^n}|u|^{2(\beta-1)}trace(YMY^H), \end{split} \end{align} | (4.8) |
where
Y = \left( \begin{array}{c} \overline{u}\nabla u \\ u\nabla \overline{u} \\ \end{array} \right)^H, M = \left( \begin{array}{cc} -\frac{\beta+1}{2} & \frac{\beta(1+\text{i}\mu)}{2} \\ \frac{\beta(1-\text{i}\mu)}{2} & -\frac{\beta+1}{2} \\ \end{array} \right), |
and Y^H is the conjugate transpose of the matrix Y . We observe that the condition \beta\leq \frac{1}{\sqrt{1+\mu^2}-1} implies that the matrix M is nonpositive definite. Hence, we obtain
\begin{align} \begin{split} 2\mathbb{E}\left[\int^t_0e^{\mu_1s}\mbox{Re}\left((1+\text{i}\mu)|u(s)|^{2\beta}u(s),\Delta u(s)\right)ds\right]\leq 0. \end{split} \end{align} | (4.9) |
Next, we estimate the terms on the right-hand side of (4.7). For the third term on the right-hand side of (4.7), applying (2.2) and Gagliardo-Nirenberg inequality, we have
\begin{align} \begin{split} &\ \ \ \ 2\mathbb{E}\left[\mbox{Re}\int^t_0e^{\mu_1s}(G(x,u(s-\rho)),-\Delta u(s))ds\right]\leq2\mathbb{E}\left[\int^t_0e^{\mu_1s}\|\nabla u(s)\|\|\nabla G(x,u(s-\rho))\|ds\right]\\& \leq\mathbb{E}\left[\int^t_0e^{\mu_1s}\|\nabla u(s)\|^2ds\right]+\mathbb{E}\left[\int^t_0e^{\mu_1s}\|\nabla G(x,u(s-\rho))\|^2ds\right]\\& \leq\mathbb{E}\left[\int^t_0e^{\mu_1s}\|\nabla u(s)\|^2ds\right]+2\mathbb{E}\left[\int^t_0e^{\mu_1s}\|\hat{h}(x)\|^2ds\right]+2\hat{a}^2\mathbb{E}\left[\int^{t}_0e^{\mu_1s}\|\nabla u(s-\rho)\|^2ds\right]\\& \leq\mathbb{E}\left[\int^t_0e^{\mu_1s}\|(\!-\!\Delta)^{\frac{\alpha+1}{2}} u(s)\|^2ds\right]\!\!+\!\!\frac{2}{\mu_1}\|\hat{h}(x)\|^2e^{\mu_1t}\\&\ \ \ \ \!\!+\!\!\frac{c}{\mu_1}\sup\limits_{s\geq 0}\mathbb{E}[\| u(s)\|^2]e^{\mu_1t}\!\!+\!\!\frac{2\hat{a}^2}{\mu_1}\sup\limits_{-\rho\leq s\leq 0}\mathbb{E}[\|\nabla \varphi(s)\|^2]e^{\mu_1t}, \end{split} \end{align} | (4.10) |
where c is a positive constant from Gagliardo-Nirenberg inequality. For the forth term on the right-hand side of (4.7), applying (2.6) and (2.7), we have
\begin{align} \begin{split} &\ \ \ \ \sum\limits^\infty_{k = 1}\mathbb{E}\left[\int^t_0e^{\mu_1s}\|\nabla(\sigma_{1,k}+\kappa\sigma_{2,k}(u(s)))\|^2ds\right] \leq2\sum\limits^\infty_{k = 1}\mathbb{E}\left[\int^t_0e^{\mu_1s}(\|\nabla\sigma_{1,k}\|^2\!\!+\!\!\|\nabla(\kappa\sigma_{2,k}(u(s)))\|^2)ds\right]\\& \leq\frac{2}{\mu_1}\sum\limits^\infty_{k = 1}\|\nabla \sigma_{1,k}\|^2e^{\mu_1t} \!\!+\!\!8\sum\limits^\infty_{k = 1}\mathbb{E}\left[\int^t_0e^{\mu_1s}\left(\beta^2_k\|\nabla \kappa\|^2\!\!+\!\!\hat{\beta}^2_k\|\kappa\|^2\!\!+\!\!{\gamma}^2_kC^2\|u(s)\|^2\!\!+\!\!\hat{\gamma}^2_k\|\kappa\|^2_{L^\infty}\|\nabla u(s)\|^2\right)ds\right]\\& \leq\frac{2}{\mu_1}\sum\limits^\infty_{k = 1}\left(\|\nabla\sigma_{1,k}\|^2+4\beta^2_k\|\nabla\kappa\|^2+4\hat{\beta}^2_k\|\kappa\|^2+4C^2{\gamma}^2_k\sup\limits_{s\geq 0}\mathbb{E}[\|u(s)\|^2]\right)e^{\mu_1t}\\&\ \ \ \ \ +8\sum\limits^\infty_{k = 1}\hat{\gamma}^2_k\|\kappa(x)\|^2_{L^\infty}\mathbb{E}\left[\int^t_0e^{\mu_1s}\|\nabla u(s)\|^2ds\right]. \end{split} \end{align} | (4.11) |
By (4.7), (4.10) and (4.11), we obtain
\begin{align} \begin{split} &\ \ \ \ \mathbb{E}[\|\nabla u(t)\|^2]\!\!+\!\!\mathbb{E}\left[\int^t_0e^{\mu_1(s-t)}\|(-\Delta)^{\frac{\alpha+1}{2}}u(s)\|^2ds\right]\\&\ \leq \mathbb{E}\left[\|\nabla\varphi(0)\|^2\right]e^{-\mu_1t}+\frac{2}{\mu_1}\|\hat{h}(x)\|^2 +\left(\mu_1-2\lambda+8\|\kappa\|^2_{L^\infty}\sum\limits^\infty_{k = 1}\hat{\gamma}^2_k\right)\mathbb{E}\left[\int^t_0e^{\mu_1s}\|\nabla u(s)\|^2ds\right]\\&\ \ \ \ +\frac{2}{\mu_1}\left(\frac c2+4\left(C^2\sum\limits^\infty_{k = 1}{\gamma}^2_k+c\|\kappa\|^2_{L^\infty}\sum\limits^\infty_{k = 1}\hat{\gamma}^2_k\right)\right)\sup\limits_{s\geq -\rho}\mathbb{E}[\|u(s)\|^2]\\& \ \ \ \ +\frac{2}{\mu_1}\sum\limits^\infty_{k = 1}\left(\|\nabla\sigma_{1,k}\|^2+4(\beta^2_k+\hat{\beta}^2_k)\|\kappa\|^2_V\right)+\frac{2\hat{a}^2}{\mu_1}\sup\limits_{-\rho\leq s\leq 0}\mathbb{E}[\|\nabla \varphi(s)\|^2]. \end{split} \end{align} | (4.12) |
Then by (4.6) and (4.12), we obtain that for all t\geq 0 ,
\begin{align} \begin{split} &\ \ \ \ \mathbb{E}[\|\nabla u(t)\|^2]+\mathbb{E}\left[\int^t_0e^{\mu_1(s-t)}\|(-\Delta)^{\frac{\alpha+1}{2}}u(s)\|^2ds\right]\\&\ \leq \mathbb{E}\left[\|\nabla\varphi(0)\|^2\right]e^{-\mu_1t}\!+\!\frac{2}{\mu_1}\|\hat{h}(x)\|^2\!+\!\frac{2}{\mu_1}\left(\frac c2\!+\!4(C^2\sum\limits^\infty_{k = 1}{\gamma}^2_k\!+\!c\|\kappa\|^2_{L^\infty}\sum\limits^\infty_{k = 1}\hat{\gamma}^2_k)\right)\sup\limits_{s\geq -\rho}\mathbb{E}[\|u(s)\|^2]\\& \ \ \ \ +\frac{2}{\mu_1}\sum\limits^\infty_{k = 1}\left(\|\nabla\sigma_{1,k}\|^2+4(\beta^2_k+\hat{\beta}^2_k)\|\kappa\|^2_V\right)+\frac{2\hat{a}^2}{\mu_1}\sup\limits_{-\rho\leq s\leq 0}\mathbb{E}[\|\nabla \varphi(s)\|^2]. \end{split} \end{align} | (4.13) |
Then by (4.13) and Lemma 3.1, we obtain the estimates (4.5).
Lemma 4.2. Suppose (2.1)–(2.7) and (4.4) hold. If \varphi(s)\in L^2(\Omega; C([-\rho, 0], V)) , then the solution u of (1.1) and (1.2) satisfies
\begin{align} \begin{split} \sup\limits_{t\geq \rho}\left\{\mathbb{E}\left[\sup\limits_{t-\rho\leq r\leq t}\|\nabla u(r)\|^2\right]\right\} \leq M_5\left(\mathbb{E}[\|\varphi\|^2_{C_V}]+1\right), \end{split} \end{align} | (4.14) |
where M_5 is a positive constant independent of \varphi .
Proof. By (1.1) and Ito's formula, we get for all t\geq \rho and t-\rho\leq r\leq t ,
\begin{align} \begin{split} &\ \ \ \ \|\nabla u(r)\|^2+2\int^r_{t-\rho}\|(-\Delta)^{\frac{\alpha+1}{2}}u(s)\|^2ds\\&\ \ \ \ +2\int^r_{t-\rho}\mbox{Re}\left((1+\text{i}\mu)|u(s)|^{2\beta}u(s),-\Delta u(s)\right)ds+2\lambda\int^r_{t-\rho}\|\nabla u(s)\|^2ds\\& = \|\nabla u(t-\rho)\|^2+2\mbox{Re}\int^r_{t-\rho}(G(x,u(s-\rho)),-\Delta u(s))ds\\& \ \ \ \ +\sum\limits^\infty_{k = 1}\int^r_{t-\rho}\|\nabla(\sigma_{1,k}+\kappa\sigma_{2,k}(u(s)))\|^2ds\\& \ \ \ \ \ +2\sum\limits^\infty_{k = 1}\mbox{Re}\int^r_{t-\rho}\left(\sigma_{1,k}(x)+\kappa(x)\sigma_{2,k}(u(s)),-\Delta u(s)\right)dW_k(s). \end{split} \end{align} | (4.15) |
For the third term on the left-hand side of (4.15), applying (4.8), we have
\begin{align} \begin{split} -2\int^r_{t-\rho}\mbox{Re}\left((1+\text{i}\mu)|u(s)|^{2\beta}u(s),-\Delta u(s)\right)ds\leq 0. \end{split} \end{align} | (4.16) |
For the second term on the right-hand side of (4.15), applying (2.2) and Gagliardo-Nirenberg inequality, we have
\begin{align} \begin{split} &\ \ \ \ 2\mbox{Re}\int^r_{t-\rho}(G(x,u(s-\rho)),-\Delta u(s))ds\leq2\int^r_{t-\rho}\|\nabla u(s)\|\|\nabla G(x,u(s-\rho))\|ds\\& \leq\int^r_{t-\rho}\|\nabla u(s)\|^2ds+\int^r_{t-\rho}\|\nabla G(x,u(s-\rho))\|^2ds\\& \leq\int^r_{t-\rho}\|\nabla u(s)\|^2ds+2\int^r_{t-\rho}\|\hat{h}(x)\|^2ds+2\hat{a}^2\int^r_{t-\rho}\|\nabla u(s-\rho)\|^2ds\\& \leq\int^r_{t-\rho}\|(\!-\!\Delta)^{\frac{\alpha+1}{2}} u(s)\|^2ds\!+\!2\rho\|\hat{h}(x)\|^2+2\hat{a}^2\int^{r-\rho}_{t-2\rho}\|\nabla u(s)\|^2ds\!+c\int^r_{t-\rho}\|u(s)\|^2ds. \end{split} \end{align} | (4.17) |
For the third term on the right-hand side of (4.15), applying (2.6) and (2.7), we have
\begin{align} \begin{split} &\ \ \ \ \sum\limits^\infty_{k = 1}\int^r_{t-\rho}\|\nabla(\sigma_{1,k}+\kappa\sigma_{2,k}(u(s)))\|^2ds \leq2\sum\limits^\infty_{k = 1}\int^r_{t-\rho}(\|\nabla\sigma_{1,k}\|^2+\|\nabla(\kappa\sigma_{2,k}(u(s)))\|^2)ds\\& \leq 2\rho\sum\limits^\infty_{k = 1}\|\nabla \sigma_{1,k}\|^2+8\rho\left(\|\nabla \kappa\|^2\sum\limits^\infty_{k = 1}\beta^2_k+\|\kappa\|^2\sum\limits^\infty_{k = 1}\hat{\beta}^2_k\right)\\&\ \ \ \ \ +8C^2\sum\limits^\infty_{k = 1}{\gamma}^2_k\int^r_{t-\rho}\|u(s)\|^2ds +8\|\kappa\|^2_{L^\infty}\sum\limits^\infty_{k = 1}\hat{\gamma}^2_k\int^r_{t-\rho}\|\nabla u(s)\|^2ds. \end{split} \end{align} | (4.18) |
By (4.4) and (4.15)–(4.18), we infer that for all t\geq \rho and t-\rho\leq r\leq t ,
\begin{align} \begin{split} &\ \ \ \ \|\nabla u(r)\|^2\leq c_1+\|\nabla u(t-\rho)\|^2+c_2\int^r_{t-2\rho}\|u(s)\|^2ds+2\hat{a}^2\int^{r}_{t-2\rho}\|\nabla u(s)\|^2ds\\& \ \ \ \ \ +2\sum\limits^\infty_{k = 1}\mbox{Re}\int^r_{t-\rho}\left(\sigma_{1,k}(x)+\kappa(x)\sigma_{2,k}(u(s)),-\Delta u(s)\right)dW_k(s), \end{split} \end{align} | (4.19) |
where c_1 and c_2 are positive constants. By (4.19), we deduce that for all t\geq \rho ,
\begin{align} \begin{split} &\ \ \ \ \mathbb{E}\left[\sup\limits_{t-\rho\leq r\leq t}\|\nabla u(r)\|^2\right]\leq c_1+\mathbb{E}\left[\|\nabla u(t-\rho)\|^2\right]+c_2\int^r_{t-2\rho}\mathbb{E}\left[\|u(s)\|^2+\|\nabla u(s)\|^2\right]ds\\& \ \ \ \ \ +2\mathbb{E}\left[\sup\limits_{t-\rho\leq r\leq t}\left|\sum\limits^\infty_{k = 1}\int^r_{t-\rho}\left(\sigma_{1,k}(x)+\kappa(x)\sigma_{2,k}(u(s)),-\Delta u(s)\right)dW_k(s)\right|\right]. \end{split} \end{align} | (4.20) |
For the second term on the right-hand side of (4.20), by Lemma 4.1 we infer that for all t\geq \rho ,
\begin{align} \begin{split} &\ \ \ \ \mathbb{E}\left[\|\nabla u(t-\rho)\|^2\right]\leq \sup\limits_{s\geq-\rho}\mathbb{E}\left[\|\nabla u(s)\|^2\right]\leq c_3\mathbb{E}\left[\|\varphi\|^2_{C_V}\right]+c_3. \end{split} \end{align} | (4.21) |
For the third term on the right-hand side of (4.20), by Lemmas 3.1 and 4.1 we infer that for all t\geq \rho ,
\begin{align} \begin{split} &\ \ \ \ c_2\int^r_{t-2\rho}\mathbb{E}\left[\|u(s)\|^2+\|\nabla u(s)\|^2\right]ds\leq 2\rho c_2 \sup\limits_{s\geq-\rho}\mathbb{E}\left[\|u(s)\|^2+\|\nabla u(s)\|^2\right]\leq c_4\mathbb{E}\left[\|\varphi\|^2_{C_V}\right]+c_4. \end{split} \end{align} | (4.22) |
For the last term on the right-hand side of (4.20), by BDG inequality, (4.18), Lemmas 3.1 and 4.1, we deduce that for all t\geq \rho ,
\begin{align} \begin{split} &\ \ \ \ 2\mathbb{E}\left[\sup\limits_{t-\rho\leq r\leq t}\left|\sum\limits^\infty_{k = 1}\int^r_{t-\rho}\left(\sigma_{1,k}(x)+\kappa(x)\sigma_{2,k}(u(s)),-\Delta u(s)\right)dW_k(s)\right|\right]\\& \leq2c_5\mathbb{E}\left[\left(\sum\limits^\infty_{k = 1}\int^t_{t-\rho}\left|\left(\sigma_{1,k}(x)+\kappa(x)\sigma_{2,k}(u(s)),-\Delta u(s)\right)\right|^2ds\right)^{\frac 12}\right]\\& \leq2c_5\mathbb{E}\left[\left(\sum\limits^\infty_{k = 1}\int^t_{t-\rho}\|\nabla u(s)\|^2\|\nabla\left(\sigma_{1,k}(x)+\kappa(x)\sigma_{2,k}(u(s))\right)\|^2ds\right)^{\frac 12}\right]\\&\leq2c_5\mathbb{E}\left[\sup\limits_{t-\rho\leq s\leq t}\|\nabla u(s)\|\left(\sum\limits^\infty_{k = 1}\int^t_{t-\rho}\|\nabla\left(\sigma_{1,k}(x)+\kappa(x)\sigma_{2,k}(u(s))\right)\|^2ds\right)^{\frac 12}\right]\\&\leq \frac 12\mathbb{E}\left[\sup\limits_{t-\rho\leq s\leq t}\|\nabla u(s)\|^2\right]+2c_5^2\mathbb{E}\left[\sum\limits^\infty_{k = 1}\int^t_{t-\rho}\|\nabla\left(\sigma_{1,k}(x)+\kappa(x)\sigma_{2,k}(u(s))\right)\|^2ds\right]\\&\leq \frac 12\mathbb{E}\left[\sup\limits_{t-\rho\leq s\leq t}\|\nabla u(s)\|^2\right]+c_6+c_6\int^t_{t-\rho}\mathbb{E}\left[\|u(s)\|^2+\|\nabla u(s)\|^2\right]ds\\&\leq \frac 12\mathbb{E}\left[\sup\limits_{t-\rho\leq s\leq t}\|\nabla u(s)\|^2\right]+c_6+\rho c_6\left(\sup\limits_{s\geq 0}\mathbb{E}\|u(s)\|^2+\sup\limits_{s\geq 0}\|\nabla u(s)\|^2\right)\\&\leq \frac 12\mathbb{E}\left[\sup\limits_{t-\rho\leq s\leq t}\|\nabla u(s)\|^2\right]+c_7\mathbb{E}\left[\|\varphi\|^2_{C_V}\right]+c_7. \end{split} \end{align} | (4.23) |
By (4.20)–(4.23), we obtain that for all t\geq \rho ,
\mathbb{E}\left[\sup\limits_{t-\rho\leq r\leq t}\|\nabla u(r)\|^2\right]\leq c_8\mathbb{E}\left[\|\varphi\|^2_{C_V}\right]+c_9, |
which completes the proof.
Lemma 4.3. Suppose (2.1)–(2.7) and (3.1) hold. If \varphi(s)\in L^{2p}(\Omega, C([-\rho, 0], V)) , then there exists a positive constant \mu_5 such that the solution u of (1.1) and (1.2) satisfies
\begin{align} \begin{split} &\ \sup\limits_{t\geq -\rho}\mathbb{E}[\|\nabla u(t)\|^{2p}]+\sup\limits_{t\geq 0}\mathbb{E}\left[\int^t_0e^{\mu_5(s-t)}\|\nabla u(s)\|^{2(p-1)}\|(-\Delta)^{\frac{\alpha+1}{2}}u(s)\|^2ds\right]\\&\ \ \ \ \ \ \ \ \leq M_5\left(\mathbb{E}\left[\|\varphi\|^{2p}_{C_V}\right]+1\right), \end{split} \end{align} | (4.24) |
where M_5 is a positive constant independent of \varphi .
Proof. By (3.1), there exist positive constants \mu and \epsilon_1 such that
\begin{align} \begin{split} &\ \mu+4(p-1)\epsilon_1^{\frac {p}{p-1}}+\frac{2p\hat{a}^{2}}{\mu\epsilon_1^p} +8C^2(p-1)(2p-1)\epsilon_1^{\frac p{p-1}}\sum\limits^\infty_{k = 1}{\gamma}^2_k +8p(2p-1)\|\kappa\|^2_{L^\infty}\sum\limits^\infty_{k = 1}\hat{\gamma}^2_k\\&\ \ \ \ \ \ \ \ \ +2(p-1)(2p-1)\epsilon_1^{\frac p{p-1}}\sum\limits^\infty_{k = 1}\left(\|\nabla\sigma_{1,k}\|^2+4\beta^2_k\|\nabla \kappa\|^2\!\!+\!\!4\hat{\beta}^2_k\|\kappa\|^2\right)\leq 2p\lambda. \end{split} \end{align} | (4.25) |
By (1.1) and applying Ito's formula to e^{\mu t}\|\nabla u(t)\|^{2p} , we get for t\geq 0 ,
\begin{align*} &\ \ \ \ e^{\mu t}\|\nabla u(t)\|^{2p}+2p\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}\|(-\Delta)^{\frac{\alpha+1}{2}}u(s)\|^2ds\\&\ \ \ \ \ \ +2p\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}\mbox{Re}\left((1+\text{i}\mu)|u(s)|^{2\beta}u(s),-\Delta u(s)\right)ds\\& = \|\nabla\varphi(0)\|^{2p}+(\mu-2p\lambda)\int^t_0e^{\mu s}\|\nabla u(s)\|^{2p}ds\\&\ \ \ \ \ +2p\mbox{Re}\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}(G(x,u(s-\rho)),-\Delta u(s))ds\\& \ \ \ \ \ +p\sum\limits^\infty_{k = 1}\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}\|\nabla(\sigma_{1,k}+\kappa\sigma_{2,k}(u(s)))\|^2ds\\& \ \ \ \ \ +2p\sum\limits^\infty_{k = 1}\mbox{Re}\int^t_0e^{\mu_1s}\|\nabla u(s)\|^{2(p-1)}\left(\sigma_{1,k}+\kappa\sigma_{2,k}(u(s)),-\Delta u(s)\right)dW_k(s)\\& \ \ \ \ \ +2p(p-1)\sum\limits^\infty_{k = 1}\mbox{Re}\int^t_0e^{\mu_1s}\|\nabla u(s)\|^{2(p-2)}\left|(\sigma_{1,k}+\kappa\sigma_{2,k}(u(s)),-\Delta u(s))\right|^2ds. \end{align*} |
Taking the expectation, we have for t\geq 0 ,
\begin{align} \begin{split} &\ \ \ \ e^{\mu t}\mathbb{E}\left[\|\nabla u(t)\|^{2p}\right]+2p\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}\|(-\Delta)^{\frac{\alpha+1}{2}}u(s)\|^2ds\right]\\&\ \ \ \ \ \ +2p\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}\mbox{Re}\left((1+\text{i}\mu)|u(s)|^{2\beta}u(s),-\Delta u(s)\right)ds\right]\\& = \mathbb{E}\left[\|\nabla\varphi(0)\|^{2p}\right]+(\mu-2p\lambda)\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2p}ds\right]\\&\ \ \ \ \ +2p\mathbb{E}\left[\mbox{Re}\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}(G(x,u(s-\rho)),-\Delta u(s))ds\right]\\& \ \ \ \ \ +p\sum\limits^\infty_{k = 1}\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}\|\nabla(\sigma_{1,k}+\kappa\sigma_{2,k}(u(s)))\|^2ds\right]\\& \ \ \ \ \ +2p(p-1)\sum\limits^\infty_{k = 1}\mathbb{E}\left[\mbox{Re}\int^t_0e^{\mu_1s}\|\nabla u(s)\|^{2(p-2)}\left|(\sigma_{1,k}+\kappa\sigma_{2,k}(u(s)),-\Delta u(s))\right|^2ds\right]. \end{split} \end{align} | (4.26) |
By (4.8), we get the third term on the left-hand side of (4.26) is nonnegative. Next, we estimate each term on the right-hand side of (4.26). For the third term on the right-hand side of (4.26), applying (2.2), Gagliardo-Nirenberg inequality and Young's inequality, we deduce
\begin{align} \begin{split} &\ \ \ \ 2p\mathbb{E}\left[\mbox{Re}\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}(G(x,u(s-\rho)),-\Delta u(s))ds\right]\\&\leq2p\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}\|\nabla u(s)\|\|\nabla G(x,u(s-\rho))\|ds\right]\\&\leq p\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}\|\nabla u(s)\|^2ds\right]+p\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}\|\nabla G(x,u(s-\rho))\|^2ds\right]\\&\leq p\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}\|\nabla u(s)\|^2ds\right]\\&\ \ \ \ +2p\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}\|\hat{h}\|^2ds\right]+2p\hat{a}^2\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}\|\nabla u(s-\rho)\|^2ds\right]\\&\leq p\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}(\|(\!-\!\Delta)^{\frac{\alpha+1}{2}} u(s)\|^2+c\|u(s)\|^2)ds\right]+\frac 2{\epsilon_1^p}\|\hat{h}(x)\|^{2p}\int^t_0e^{\mu s}ds\\&\ \ \ \ +4(p-1)\epsilon_1^{\frac {p}{p-1}}\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2p}ds\right] +\frac{2p\hat{a}^{2}}{\epsilon_1^p}\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2p}ds\right]\\&\leq p\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}\|(\!-\!\Delta)^{\frac{\alpha+1}{2}} u(s)\|^2ds\right]+\left(4(p-1)\epsilon_1^{\frac {p}{p-1}}+\frac{2p\hat{a}^{2}}{\mu\epsilon_1^p}\right)\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2p}ds\right]\\&\ \ \ \ +\frac 1{\mu\epsilon_1^p}\|\hat{h}(x)\|^{2p}e^{\mu t} +\frac{2p\hat{a}^{2}}{\mu\epsilon_1^p}\sup\limits_{-\rho\leq s\leq 0}\mathbb{E}\left[\|\nabla \varphi(s)\|^{2p}\right]e^{\mu t}+c\sup\limits_{s\geq 0}\mathbb{E}\left[\| u(s)\|^{2p}\right]e^{\mu t}. \end{split} \end{align} | (4.27) |
For the forth term on the right-hand side of (4.26), applying (2.7), we infer
\begin{align} \begin{split} &\ \ \ \ p\sum\limits^\infty_{k = 1}\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}\|\nabla(\sigma_{1,k}+\kappa\sigma_{2,k}(u(s)))\|^2ds\right]\\& \leq2p\sum\limits^\infty_{k = 1}\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p\!-\!1)}\|\nabla\sigma_{1,k}\|^2ds\right] \!+\!2p\sum\limits^\infty_{k = 1}\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p\!-\!1)}\|\nabla(\kappa\sigma_{2,k}(u(s)))\|^2ds\right]\\&\leq 2p\sum\limits^\infty_{k = 1}\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}\|\nabla\sigma_{1,k}\|^2ds\right]\\&\ \ \ \ +8p\sum\limits^\infty_{k = 1}\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}\left(\beta^2_k\|\nabla \kappa\|^2\!\!+\!\!\hat{\beta}^2_k\|\kappa\|^2\!\!+\!\!{\gamma}^2_kC^2\|u(s)\|^2\!\!+\!\!\hat{\gamma}^2_k\|\kappa\|^2_{L^\infty}\|\nabla u(s)\|^2\right)ds\right]\\&\leq2p\sum\limits^\infty_{k = 1}\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}\left(\|\nabla\sigma_{1,k}\|^2+4\beta^2_k\|\nabla \kappa\|^2\!\!+\!\!4\hat{\beta}^2_k\|\kappa\|^2\right)ds\right]\\&\ \ \ \ +8C^2p\sum\limits^\infty_{k = 1}{\gamma}^2_k\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}\|u(s)\|^2ds\right] +8p\|\kappa\|^2_{L^\infty}\sum\limits^\infty_{k = 1}\hat{\gamma}^2_k\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2p}ds\right]. \end{split} \end{align} | (4.28) |
Then applying Young's inequality, (4.28) can be estimated by
\begin{align} \begin{split} &\ \ \ \ p\sum\limits^\infty_{k = 1}\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2(p-1)}\|\nabla(\sigma_{1,k}+\kappa\sigma_{2,k}(u(s)))\|^2ds\right]\\& \leq\sum\limits^\infty_{k = 1}\left(\|\nabla\sigma_{1,k}\|^2+4\beta^2_k\|\nabla \kappa\|^2\!\!+\!\!4\hat{\beta}^2_k\|\kappa\|^2\right)\times\mathbb{E}\left[\int^t_0e^{\mu s}\left((2p-2)\epsilon_1^{\frac p{p-1}}\|\nabla u(s)\|^{2p}+\frac2{\epsilon_1^p}\right)ds\right]\\&\ \ \ \ +2C^2\sum\limits^\infty_{k = 1}{\gamma}^2_k\mathbb{E}\left[\int^t_0e^{\mu s}\left((4p-4)\epsilon_1^{\frac p{p-1}}\|\nabla u(s)\|^{2p}+\frac4{\epsilon_1^p}\|u(s)\|^{2p}\right)ds\right]\\&\ \ \ \ +8p\|\kappa\|^2_{L^\infty}\sum\limits^\infty_{k = 1}\hat{\gamma}^2_k\mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2p}ds\right]\\& = \left[(2p-2)\epsilon_1^{\frac p{p-1}}\sum\limits^\infty_{k = 1}\left(\|\nabla\sigma_{1,k}\|^2+4\beta^2_k\|\nabla \kappa\|^2\!\!+\!\!4\hat{\beta}^2_k\|\kappa\|^2\right)\right.\\&\ \ \ \ \left.+2C^2(4p-4)\epsilon_1^{\frac p{p-1}}\sum\limits^\infty_{k = 1}{\gamma}^2_k+8p\|\kappa\|^2_{L^\infty}\sum\limits^\infty_{k = 1}\hat{\gamma}^2_k\right] \mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2p}ds\right]\\&\ \ \ \ \!+\!\sum\limits^\infty_{k = 1}\left(\|\nabla\sigma_{1,k}\|^2\!+\!4\beta^2_k\|\nabla \kappa\|^2\!\!+\!\!4\hat{\beta}^2_k\|\kappa\|^2\right)\frac2{\epsilon_1^p}e^{\mu t} \!+\!\frac{8C^2}{\mu\epsilon_1^p}\sum\limits^\infty_{k = 1}{\gamma}^2_k\sup\limits_{s\geq 0}\mathbb{E}\left[\|u(s)\|^{2p}\right]e^{\mu t}. \end{split} \end{align} | (4.29) |
For the fifth term on the right-hand side of (4.26), applying integrating by parts and (4.29), we get
\begin{align} \begin{split} &\ \ \ \ 2p(p-1)\sum\limits^\infty_{k = 1}\mathbb{E}\left[\mbox{Re}\int^t_0e^{\mu_1s}\|\nabla u(s)\|^{2(p-2)}\left|(\sigma_{1,k}+\kappa\sigma_{2,k}(u(s)),-\Delta u(s))\right|^2ds\right]\\&\leq2p(p-1)\sum\limits^\infty_{k = 1}\mathbb{E}\left[\int^t_0e^{\mu_1s}\|\nabla u(s)\|^{2p-2}\|\nabla(\sigma_{1,k}+\kappa\sigma_{2,k}(u(s)))\|^2ds\right]\\&\leq2(p-1)\left[(2p-2)\epsilon_1^{\frac p{p-1}}\sum\limits^\infty_{k = 1}\left(\|\nabla\sigma_{1,k}\|^2+4\beta^2_k\|\nabla \kappa\|^2\!\!+\!\!4\hat{\beta}^2_k\|\kappa\|^2\right)\right.\\&\ \ \ \ \left.+2C^2(4p-4)\epsilon_1^{\frac p{p-1}}\sum\limits^\infty_{k = 1}{\gamma}^2_k+8p\|\kappa\|^2_{L^\infty}\sum\limits^\infty_{k = 1}\hat{\gamma}^2_k\right] \mathbb{E}\left[\int^t_0e^{\mu s}\|\nabla u(s)\|^{2p}ds\right]\\&\ \ \ \ \ \!+\!\sum\limits^\infty_{k = 1}\left(\|\nabla\sigma_{1,k}\|^2\!+\!4\beta^2_k\|\nabla \kappa\|^2\!\!+\!\!4\hat{\beta}^2_k\|\kappa\|^2\right)\frac{4(p\!-\!1)}{\epsilon_1^p}e^{\mu t} \!+\!\frac{16C^2(p\!-\!1)}{\mu\epsilon_1^p}\sum\limits^\infty_{k = 1}{\gamma}^2_k\sup\limits_{s\geq 0}\mathbb{E}\left[\|u(s)\|^{2p}\right]e^{\mu t}. \end{split} \end{align} | (4.30) |
By (4.26), (4.27), (4.29) and (4.30), we obtain
\begin{align} \begin{split} &\ \ \ \ \mathbb{E}\left[\|\nabla u(t)\|^{2p}\right]+p\mathbb{E}\left[\int^t_0e^{\mu (s-t)}\|\nabla u(s)\|^{2(p-1)}\|(-\Delta)^{\frac{\alpha+1}{2}}u(s)\|^2ds\right]\\& \leq\mathbb{E}\left[\|\nabla\varphi(0)\|^{2p}\right]e^{-\mu t}+\left[\mu-2p\lambda+4(\varrho-1)\epsilon_1^{\frac {p}{p-1}}+\frac{2p\hat{a}^{2}}{\mu\epsilon_1^p} +8C^2(p-1)(2p-1)\epsilon_1^{\frac p{p-1}}\sum\limits^\infty_{k = 1}{\gamma}^2_k\right.\\&\ \ \ \ \left.+8p(2p-1)\|\kappa\|^2_{L^\infty}\sum\limits^\infty_{k = 1}\hat{\gamma}^2_k+2(p-1)(2p-1)\epsilon_1^{\frac p{p-1}}\sum\limits^\infty_{k = 1}\left(\|\nabla\sigma_{1,k}\|^2+4\beta^2_k\|\nabla \kappa\|^2\!\!+\!\!4\hat{\beta}^2_k\|\kappa\|^2\right)\right]\\&\ \ \ \ \times\mathbb{E}\left[\int^t_0e^{\mu (s-t)}\|\nabla u(s)\|^{2p}ds\right]+\frac 1{\mu\epsilon_1^p}\|\hat{h}(x)\|^{2p} +\frac{2p\hat{a}^{2}}{\mu\epsilon_1^p}\sup\limits_{-\rho\leq s\leq 0}\mathbb{E}\left[\|\nabla \varphi(s)\|^{2p}\right]\\&\ \ \ \ +c\sup\limits_{s\geq 0}\mathbb{E}\left[\| u(s)\|^{2p}\right] +\sum\limits^\infty_{k = 1}\left(\|\nabla\sigma_{1,k}\|^2+4\beta^2_k\|\nabla \kappa\|^2\!\!+\!\!4\hat{\beta}^2_k\|\kappa\|^2\right)\frac{4p-2}{\epsilon_1^p}\\&\ \ \ \ +\frac{8C^2(2p-1)}{\mu\epsilon_1^p}\sum\limits^\infty_{k = 1}{\gamma}^2_k\sup\limits_{s\geq 0}\mathbb{E}\left[\|u(s)\|^{2p}\right]. \end{split} \end{align} | (4.31) |
Then by (4.25) and (4.31), we deduce that for all t\geq 0 ,
\begin{align} \begin{split} &\ \ \ \ \mathbb{E}\left[\|\nabla u(t)\|^{2p}\right]+p\mathbb{E}\left[\int^t_0e^{\mu (s-t)}\|\nabla u(s)\|^{2(p-1)}\|(-\Delta)^{\frac{\alpha+1}{2}}u(s)\|^2ds\right]\\& \leq\mathbb{E}\left[\|\nabla\varphi(0)\|^{2p}\right]e^{-\mu t}+\frac 1{\mu\epsilon_1^p}\|\hat{h}(x)\|^{2p} +\frac{2p\hat{a}^{2}}{\mu\epsilon_1^p}\sup\limits_{-\rho\leq s\leq 0}\mathbb{E}\left[\|\nabla \varphi(s)\|^{2p}\right]\\&\ \ \ \ +c\sup\limits_{s\geq 0}\mathbb{E}\left[\| u(s)\|^{2p}\right] +\sum\limits^\infty_{k = 1}\left(\|\nabla\sigma_{1,k}\|^2+4\beta^2_k\|\nabla \kappa\|^2\!\!+\!\!4\hat{\beta}^2_k\|\kappa\|^2\right)\frac{4p-2}{\epsilon_1^p}\\&\ \ \ \ +\frac{8C^2(2p-1)}{\mu\epsilon_1^p}\sum\limits^\infty_{k = 1}{\gamma}^2_k\sup\limits_{s\geq 0}\mathbb{E}\left[\|u(s)\|^{2p}\right]. \end{split} \end{align} | (4.32) |
Therefore, by (4.32) and Lemma 3.5, there exists a constant M_5 independent of \varphi such that
\begin{align} \begin{split} &\ \ \ \ \sup\limits_{t\geq -\rho}\mathbb{E}\left[\|\nabla u(t)\|^{2p}\right]+\sup\limits_{t\geq 0}\mathbb{E}\left[\int^t_0e^{\mu (s-t)}\|\nabla u(s)\|^{2(p-1)}\|(-\Delta)^{\frac{\alpha+1}{2}}u(s)\|^2ds\right]\\& \leq M_5(\mathbb{E}\left[\|\varphi\|^{2p}_{C_V}\right]+1). \end{split} \end{align} | (4.33) |
For convenience, we write A = (1+\text{i}\nu)(-\triangle)^\alpha+\lambda I . Then, similar to Theorem 6.5 in [31], the solution of (1.1) and (1.2) can be expressed as
\begin{align} \begin{split} &\ u(t) = e^{-At}u(0)-\int_0^te^{-A(t-s)}(1+\text{i}\mu)|u(s)|^{2\beta}u(s)ds\\&\ \ \ +\int_0^te^{-A(t-s)}G(\cdot,u(s-\rho))ds+\sum\limits^\infty_{k = 1}\int_0^te^{-A(t-s)}(\sigma_{1,k}+\kappa\sigma_{2,k}(u(s)))dW_k(s). \end{split} \end{align} | (4.34) |
The next lemma is concerned with the H \ddot{o} lder continuity ofsolutions in time which is needed to prove the tightness of distributions of solutions.
Lemma 4.4. Suppose (2.1)–(2.7) and (3.1) hold. If \varphi(s)\in L^{2p}(\Omega, C([-\rho, 0], V)) , then the solution u of (1.1) and (1.2) satisfies, for any t > r\geq 0 ,
\begin{align} \begin{split} \mathbb{E}[\|u(t)-u(r)\|^{2p}]\leq M_6(|t-r|^{p}+|t-r|^{2p}), \end{split} \end{align} | (4.35) |
where M_6 is a positive constant depending on \varphi , but independent of t and r .
Proof. By (4.34), we get for t > r\geq 0 ,
\begin{align} \begin{split} &\ u(t) = e^{-A(t-r)}u(r)-\int_r^te^{-A(t-s)}(1+\text{i}\mu)|u(s)|^{2\beta}u(s)ds\\&\ \ \ +\int_r^te^{-A(t-s)}G(\cdot,u(s-\rho))ds+\sum\limits^\infty_{k = 1}\int_r^te^{-A(t-s)}(\sigma_{1,k}+\kappa\sigma_{2,k}(u(s)))dW_k(s). \end{split} \end{align} | (4.36) |
Then we infer
\begin{align} \begin{split} &\ \|u(t)-u(r)\|^{2p}\leq \frac{5^{2p}}{4}\left[\|(e^{-A(t-r)}-I)u(r)\|^{2p}+\|\int_r^te^{-A(t-s)}(1+\text{i}\mu)|u(s)|^{2\beta}u(s)ds\|^{2p}\right.\\&\ \ \ \left.+\|\int_r^te^{-A(t-s)}G(\cdot,u(s-\rho))ds\|^{2p}+\|\sum\limits^\infty_{k = 1}\int_r^te^{-A(t-s)}(\sigma_{1,k}+\kappa\sigma_{2,k}(u(s)))dW_k(s)\|^{2p}\right]. \end{split} \end{align} | (4.37) |
Taking the expectation of (4.36), we have for all t > r\geq 0 ,
\begin{align} \begin{split} &\ \mathbb{E}[\|u(t)-u(r)\|^{2p}]\leq \frac{5^{2p}}{4}\mathbb{E}[\|(e^{-A(t-r)}\!-\!I)u(r)\|^{2p}]\!+\!\frac{5^{2p}}{4}\mathbb{E}\left[\|\int_r^te^{-A(t-s)}(1\!+\!\text{i}\mu)|u(s)|^{2\beta}u(s)ds\|^{2p}\right]\\&\ \ \ \!+\!\frac{5^{2p}}{4}\mathbb{E}\left[\|\int_r^te^{-A(t-s)}G(\cdot,u(s\!-\!\rho))ds\|^{2p}\right] \!+\!\frac{5^{2p}}{4}\mathbb{E}\left[\|\sum\limits^\infty_{k = 1}\int_r^te^{-A(t-s)}(\sigma_{1,k}\!+\!\kappa\sigma_{2,k}(u(s)))dW_k(s)\|^{2p}\right]. \end{split} \end{align} | (4.38) |
For the first term on the right-hand side of (4.38), by Theorem 1.4.3 in [32], we find that there exists a positive number C_0 depending on \varrho such that for all t > r\geq 0 ,
\begin{align*} \frac{5^{2p}}{4}\mathbb{E}[\|(e^{-A(t-r)}\!-\!I)u(r)\|^{2p}]\leq C_0(t-r)^p\mathbb{E}[\|u(r)\|^{2p}_{C_V}]. \end{align*} |
Applying Lemmas 3.5 and 4.3, we obtain for all t > r\geq 0 ,
\begin{align} \begin{split} \frac{5^{2p}}{4}\mathbb{E}[\|(e^{-A(t-r)}\!-\!I)u(r)\|^{2p}]\leq C_1(t-r)^p. \end{split} \end{align} | (4.39) |
For the second term on the right-hand side of (4.38), by the contraction property of e^{-At} , we infer that for all t > r\geq 0 ,
\begin{align*} &\ \ \ \mathbb{E}[\|\int_r^te^{-A(t-s)}(1\!+\!\text{i}\mu)|u(s)|^{2\beta}u(s)ds\|^{2p}] \leq(1\!+\!\mu^2)^p\mathbb{E}\left[\left(\int_r^t\||u(s)|^{2\beta+1}\|ds\right)^{2p}\right]\\& \leq (1\!+\!\mu^2)^p\mathbb{E}\left[\left(\int_r^t\||u(s)|^{2\beta+1}\|^{2p}ds\right)\right](t-r)^{2p-1}\\& \leq(1\!+\!\mu^2)^p\sup\limits_{s\geq 0}\mathbb{E}\left[\left(\|u(s)\|^{2(2\beta+1)}_{L^{2(2\beta+1)}}\right)^{p}\right](t-r)^{2p}. \end{align*} |
We deduce the estimate \sup\limits_{s\geq 0}\mathbb{E}\left[\left(\|u(s)\|^{2(2\beta+1)}_{L^{2(2\beta+1)}}\right)^{p}\right]\leq M'\left(\mathbb{E}[\|\varphi\|^2_{C_V}]+1\right) similarly to Lemma 3.5 together with Lemma 3.3 in [1]. Hence, the second term on the right-hand side of (4.38) can be estimated by
\begin{align} \begin{split} \mathbb{E}\left[\|\int_r^te^{-A(t-s)}(1\!+\!\text{i}\mu)|u(s)|^{2\beta}u(s)ds\|^{2p}\right] \leq C_2(t-r)^{2p}. \end{split} \end{align} | (4.40) |
For the third term on the right-hand side of (4.38), by the contraction property of e^{-At} and (2.1) and Lemma 3.5, we deduce that for all t > r\geq 0 ,
\begin{align} \begin{split} &\ \ \ \ \frac{5^{2p}}{4}\mathbb{E}\left[\|\int_r^te^{-A(t-s)}G(\cdot,u(s\!-\!\rho))ds\|^{2p}\right] \leq \frac{5^{2p}}{4}\mathbb{E}\left[\left(\int_r^t\|G(\cdot,u(s\!-\!\rho))\|ds\right)^{2p}\right]\\& \leq\frac{5^{2p}}{4}\mathbb{E}\left[\left(\int_r^t(\|h\|+a\|u(s\!-\!\rho)\|)ds\right)^{2p}\right]\\& \leq\frac{5^{2p}}{4}\mathbb{E}\left[\left(\int_r^t(\|h\|+a\|u(s\!-\!\rho)\|)^{2p}ds\right)\right](t-r)^{2p-1}\\& \leq\frac{10^{2p}}{8}(t-r)^{2p-1}\int_r^t\left(\|h\|^{2p}+a^{2p}\mathbb{E}\left[\|u(s\!-\!\rho)\|^{2p}\right]\right)ds\\& \leq\frac{10^{2p}}{8}\left(\|h\|^{2p}+a^{2p}\sup\limits_{t\geq -\rho}\mathbb{E}\left[\|u(s)\|^{2p}\right]\right)(t-r)^{2p}\leq C_3(t-r)^{2p}. \end{split} \end{align} | (4.41) |
For the forth term the right-hand side of (4.38), from the BDG inequality, the contraction property of e^{-At} , (2.6) H \ddot{o} lder's inequality and Lemma 3.5, we deduce
\begin{align*} &\frac{5^{2p}}{4}\mathbb{E}\left[\|\sum\limits^\infty_{k = 1}\int_r^te^{-A(t-s)}(\sigma_{1,k}\!+\!\kappa\sigma_{2,k}(u(s)))dW_k(s)\|^{2p}\right]\\ &\leq\frac{5^{2p}}{4}C_4\mathbb{E}\left[\left(\int_r^t\sum\limits^\infty_{k = 1}\|e^{-A(t-s)}(\sigma_{1,k}\!+\!\kappa\sigma_{2,k}(u(s)))\|^2ds\right)^{p}\right]\\ &\leq\frac{5^{2p}}{4}C_4\mathbb{E}\left[\left(\int_r^t\sum\limits^\infty_{k = 1}2(\|\sigma_{1,k}\|^2+\|\kappa\sigma_{2,k}(u(s))\|^2)ds\right)^{p}\right]\\ &\leq\frac{5^{2p}}{4}C_4\mathbb{E}\left[\left(\int_r^t\sum\limits^\infty_{k = 1}2 (\|\sigma_{1,k}\|^2+2\|\kappa\|^2\beta^2_k+2\|\kappa\|^2_{L^\infty}\gamma^2_k\|(u(s))\|^2)ds\right)^{p}\right] \\ &\leq\frac{5^{2p}}{2}C_4\mathbb{E}\left[\left(2\sum\limits^\infty_{k = 1}(\|\sigma_{1,k}\|^2+2\|\kappa\|^2\beta^2_k)(t-r) +4\sum\limits^\infty_{k = 1}\|\kappa\|^2_{L^\infty}\gamma^2_k\int_r^t\|u(s)\|^2ds\right)^{p}\right]\\& \leq\frac{10^{2p}}{8}C_4\left(\sum\limits^\infty_{k = 1}(\|\sigma_{1,k}\|^2+2\|\kappa\|^2\beta^2_k\right)^p(t-r)^p +\frac{10^{2p}}{8}C_4\left(2\sum\limits^\infty_{k = 1}\|\kappa\|^2_{L^\infty}\gamma^2_k\right)^p\mathbb{E}\left[\left(\int_r^t\|u(s)\|^2ds\right)^{p}\right]\\& \leq\frac{10^{2p}}{8}C_4\left(\sum\limits^\infty_{k = 1}(\|\sigma_{1,k}\|^2+2\|\kappa\|^2\beta^2_k\right)^p(t-r)^p\\&\ \ \ \ +\frac{10^{2p}}{8}C_4\left(2\sum\limits^\infty_{k = 1}\|\kappa\|^2_{L^\infty}\gamma^2_k\right)^p(t-r)^{p-1} \int_r^t\mathbb{E}\left[\|u(s)\|^{2p}\right]ds\\& \leq\frac{10^{2p}}{8}C_4\left(\sum\limits^\infty_{k = 1}(\|\sigma_{1,k}\|^2+2\|\kappa\|^2\beta^2_k\right)^p(t-r)^p\\&\ \ \ \ +\frac{10^{2p}}{8}C_4\left(2\sum\limits^\infty_{k = 1}\|\kappa\|^2_{L^\infty}\gamma^2_k\right)^p(t-r)^{p} \sup\limits_{s\geq 0}\mathbb{E}\left[\|u(s)\|^{2p}\right]\\&\leq C_5(t-r)^p. \end{align*} | (4.42) |
Therefore, from (4.38)–(4.42), we obtain there exists C_6 > 0 independent of t and r , such that for all t > r\geq 0 ,
\mathbb{E}[\|u(t)-u(r)\|^{2p}]\leq C_6(|t-r|^{p}+|t-r|^{2p}). |
The proof is complete.
In this section, we first recall the definition of invariant measure and transition operator. Then we construct a compact subset of C([-\rho, 0];H) in order to prove the tightness of the sequence of invariant measure m_k on C([-\rho, 0];H) .
Recall that for any initial time t_0 and every \mathcal {F}_{t_0} -measurable function \varphi(s)\in L^2(\Omega, C([-\rho, 0], H)) , problems (1.1) and (1.2) has a unique solution u(t; t_0, \varphi) for t\in[t_0-\rho, \infty) . For convenience, given t\geq t_0 and \mathcal {F}_{t_0} -measurable function \varphi(s)\in L^2(\Omega, C([-\rho, 0], H)) , the segment of u(t; t_0, \varphi) on [t-\rho, t] is written as
u_t(t_0,\varphi)(s) = u(t+s;t_0,\varphi)\ for\ every\ s\in[-\rho,0]. |
Then u_t(t_0, \varphi)\in L^2(\Omega, C([-\rho, 0], H)) for all t\geq t_0 . We introduce the transition operator for (1.1). If \phi(s):C([-\rho, 0], H)\rightarrow \mathbb{R} is a bounded Borel function, then for initial time r with 0\leq r\leq t and \varphi(s)\in C([-\rho, 0], H) , we write
(p_{r,t}\phi)(\varphi) = \mathbb{E}[\phi(u_t(r,\varphi))]. |
Particularly, for \Gamma\in \mathcal {B}(C([-\rho, 0], H)) , 0\leq r\leq t and \varphi\in C([-\rho, 0], H) , we have
p(r,\varphi;t,\Gamma) = (p_{r,t}1_{\Gamma})(\varphi) = P\{\omega\in\Omega|u_t(r,\varphi)\in\Gamma\}, |
where 1_{\Gamma} is the characteristic function of \Gamma . Then p(r, \varphi; t, \cdot) is the distribution of u_t(0, \varphi) in C([-\rho, 0], H) . In the following context, we will write p_{0, t} as p_t .
Recall that a probability measure \mathscr{M} on C([-\rho, 0], H) is called an invariant measure, if for all t\geq0 and every bounded and continuous function \phi:C([-\rho, 0];H)\rightarrow \mathbb{R},
\int_{C([-\rho,0];H)}(p_t\phi)(\varphi)d\mathscr{M}(\varphi) = \int_{C([-\rho,0];H)}\phi(\varphi)d\mathscr{M}(\varphi),\ \ for\ all\ t\geq0. |
According to [33], we infer that the transition operator \{p_{r, t}\}_{0\leq r\leq t} has the following properties.
Lemma 5.1. Suppose (2.1)–(2.7) and (4.1)–(4.3) hold. One has
(a) The family \{p_{r, t}\}_{0\leq r\leq t} is Feller; that is, if \phi:C([-\rho, 0], H)\rightarrow \mathbb{R} is bounded and continuous, then for any 0\leq r\leq t , the function p_{r, t}\phi:C([-\rho, 0], H)\rightarrow \mathbb{R} is also bounded and continuous.
(b) The family \{p_{r, t}\}_{0\leq r\leq t} is homogeneous (in time); that is, for any 0\leq r\leq t ,
p(r,\varphi;t,\cdot) = p(0,\varphi;t-r,\cdot), \forall\varphi\in C([-\rho,0],H). |
(c) Given r\geq0 and \varphi\in C([-\rho, 0], H) , the process \{u_t(r, \varphi)\}_{t\geq r} is a C([-\rho, 0], H) -valued Markov process. Consequently, if \phi:C([-\rho, 0], H)\rightarrow \mathbb{R} is a bounded Borel function, then for any 0\leq s\leq r\leq t , P -almost surely,
(p_{s,t}\phi)(\varphi) = (p_{s,r}(p_{r,t}\phi))(\varphi), \forall\varphi\in C([-\rho,0],H), |
and the Chapman-Kolmogorov equation is valid:
p(s,\varphi;t,\Gamma) = \int_{C([-\rho,0],H)}p(s,\varphi;r,dy)p(r,y;t,\Gamma), |
for any \varphi\in C([-\rho, 0], H) and \Gamma\in\mathcal {B}(C([-\rho, 0], H)).
Now, we establish the existence of invariant measures of problems (1.1) and (1.2).
Theorem 5.2. Suppose (2.1)–(2.7) and (4.1)–(4.3) hold. Then (1.1) and (1.2) processes an invariant measure on C([-\rho, 0], H) .
Proof. We employ Krylov-Bogolyubov's method to the solution u(t, 0, 0) of problems (1.1) and (1.2), where the initial condition \varphi\equiv0 at the initial time 0. Because of this particular \varphi\in C([-\rho, 0], V)\subseteq C([-\rho, 0], H) , we know that all results obtained in the previous Sections 3 and 4 are valid. For simplicity, the solution u(t, 0, 0) is written as u(t) and the segment u_t(0, 0) as u_t . For k\in\mathbb{N}^+ , we set
\begin{align} \mathscr{M}_k = \frac1k\int^{k+\rho}_{\rho} p(0,0;t,\cdot)dt. \end{align} | (5.1) |
Step 1. We prove the tightness of \{\mathscr{M}_k\}_{k = 1}^\infty in C([-\rho, 0], H) . Applying Lemmas 3.2 and 4.2, we get that there exists C_1 > 0 such that for all t\geq \rho ,
\begin{align} \mathbb{E}\left[\sup\limits_{-\rho\leq s\leq 0}\|u_t(s)\|^{2}_{V}\right]\leq C_1. \end{align} | (5.2) |
By (5.2) and Chebyshev's inequality, we have that for all t\geq \rho ,
\begin{align*} P\left(\left\{\sup\limits_{-\rho\leq s\leq 0}\|u_t(s)\|_{V}\geq R\right\}\right)\leq\frac 1{R^2}\mathbb{E}\left[\sup\limits_{-\rho\leq s\leq 0}\|u_t(s)\|^2_{V}\right]\leq\frac{C_1}{R^2}\rightarrow 0\ \ as\ \ R\rightarrow \infty, \end{align*} |
and hence for every \varepsilon > 0 , there exists R_1 = R_1(\varepsilon) > 0 such that for all t\geq \rho ,
\begin{align} P\left(\left\{\sup\limits_{-\rho\leq s\leq 0}\|u_t(s)\|_{V}\geq R_1\right\}\right)\leq \frac 13\varepsilon. \end{align} | (5.3) |
By Lemma 4.4, we get that there exists C_2 > 0 such that for all t\geq \rho and r, s\in[-\rho, 0] ,
\mathbb{E}[\|u_t(r)-u_t(s)\|^{2p}]\leq C_2(1+|r-s|^{p})|r-s|^{p}, |
and hence for all t\geq \rho and r, s\in[-\rho, 0] ,
\begin{align} \mathbb{E}[\|u_t(r)-u_t(s)\|^{2p}]\leq C_2(1+\rho^{p})|r-s|^{p}. \end{align} | (5.4) |
Since p\geq 2 , applying (5.4) and the usual technique of dyadic division, we obtain that there exists R_2 = R_2(\varepsilon) > 0 such that for all t\geq \rho ,
\begin{align} P\left(\left\{\sup\limits_{-\rho\leq s\leq r\leq 0}\frac{\|u_t(r)-u_t(s)\|}{|r-s|^{\frac{p-1}{4p}}} \leq R_2\right\}\right)\geq1-\frac 13\varepsilon. \end{align} | (5.5) |
By Lemma 3.4, we get that for given \varepsilon > 0 and m\in\mathbb{N} , there exists an integer n_m = n_m(\varepsilon, m)\geq 1 such that for all t\geq \rho ,
\mathbb{E}\left[\sup\limits_{-\rho\leq s\leq 0}\int_{|x|\geq n_m}|u(t+s,x)|^{2}dx\right]\leq \frac{\varepsilon}{2^{2m+2}}, |
which implies that for all t\geq \rho and m\in\mathbb{N} ,
\begin{align} P\left(\left\{\sup\limits_{-\rho\leq s\leq 0}\int_{|x|\geq n_m}|u(t+s,x)|^{2}dx \geq \frac 1{2^m}\right\}\right)\leq 2^m\mathbb{E}\left[\sup\limits_{-\rho\leq s\leq 0}\int_{|x|\geq n_m}|u(t+s,x)|^{2}dx\right]\leq \frac{\varepsilon}{2^{m+2}}. \end{align} | (5.6) |
By (5.6), we infer that for all t\geq \rho ,
P\left(\bigcup\limits_{m = 1}^{\infty}\left\{\sup\limits_{-\rho\leq s\leq 0}\int_{|x|\geq n_m}|u(t+s,x)|^{2}dx \geq \frac 1{2^m}\right\}\right)\leq \sum\limits^\infty_{k = 1}\frac{\varepsilon}{2^{m+2}}\leq \frac14\varepsilon, |
and hence for all t\geq \rho ,
\begin{align} P\left(\left\{\sup\limits_{-\rho\leq s\leq 0}\int_{|x|\geq n_m}|u(t+s,x)|^{2}dx \leq \frac 1{2^m}\ for \ all\ m\in\mathbb{N}\right\}\right)\geq 1-\frac14\varepsilon. \end{align} | (5.7) |
Let
\begin{align} \mathcal {M}_{1,\varepsilon} = \left\{\zeta:[-\rho,0]\rightarrow V,\sup\limits_{-\rho\leq s\leq 0}\|\zeta(s)\|_{V}\leq R_1(\varepsilon)\right\}, \end{align} | (5.8) |
\begin{align} \mathcal {M}_{2,\varepsilon} = \left\{\zeta\in C([-\rho,0],H):\sup\limits_{-\rho\leq s\leq r\leq 0}\frac{\|\zeta(r)-\zeta(s)\|}{|r-s|^{\frac{\varrho-1}{4\varrho}}} \leq R_2(\varepsilon)\right\}, \end{align} | (5.9) |
\begin{align} \mathcal {M}_{3,\varepsilon} = \left\{\zeta\in C([-\rho,0],H):\sup\limits_{-\rho\leq s\leq 0}\int_{|x|\geq n_m}|\zeta(s,x)|^{2}dx \leq \frac 1{2^m}\ for \ all\ m\in\mathbb{N}\right\}, \end{align} | (5.10) |
and
\begin{align} \mathcal {M}_{\varepsilon} = \mathcal {M}_{1,\varepsilon}\bigcap\mathcal {M}_{2,\varepsilon}\bigcap\mathcal {M}_{3,\varepsilon}. \end{align} | (5.11) |
From (5.3), (5.5) and (5.7)–(5.11), we obtain that for all t\geq \rho ,
\begin{align} P\left(u_t\in\mathcal {M}_{\varepsilon}\right) > 1-\varepsilon. \end{align} | (5.12) |
By (5.1) and (5.12), we deduce that for all k\in\mathbb{N} ,
\begin{align} \mathscr{M}_k\left(\mathcal {M}_{\varepsilon}\right) > 1-\varepsilon. \end{align} | (5.13) |
Next, we prove the set \mathcal {M}_{\varepsilon} is precompact in C([-\rho, 0], H) . First, we prove for every s\in[-\rho, 0] the set \{\zeta(s):\zeta\in\mathcal {M}_{\varepsilon}\} is a precompact subset of H . By (5.8) and (5.11), we obtain that for every s\in[-\rho, 0] , the set \{\zeta(s):\zeta\in\mathcal {M}_{\varepsilon}\} is bounded in V . Let \mathcal {Q}_{m_0} = \left\{x\in\mathbb{R}^n:|x| < n_{m_0}\right\} . Then we get that the set \{\zeta(s)|_{\mathcal {Q}_{m_0}}:\zeta\in\mathcal {M}_{\varepsilon}\} is bounded in H^1(\mathcal {Q}_{m_0}) and hence precompact in L^2(\mathcal {Q}_{m_0}) due to compactness of the embedding H^1(\mathcal {Q}_{m_0})\hookrightarrow L^2(\mathcal {Q}_{m_0}) . This implies that the set \{\zeta(s)|_{\mathcal {Q}_{m_0}}:\zeta\in\mathcal {M}_{\varepsilon}\} has a finite open cover of balls with radius \frac 12\delta in L^2(\mathcal {Q}_{m_0}) . Note that for every \delta > 0 , there exists m_0 = m_0(\delta)\in\mathbb{N} such that for all \zeta\in\mathcal {M}_{\varepsilon} ,
\begin{align} \int_{|x|\geq n_{m_0}}|\zeta(s,x)|^{2}dx \leq \frac 1{2^{m_0}} < \frac{\delta^2}8. \end{align} | (5.14) |
Hence, by (5.14), the set \{\zeta(s):\zeta\in\mathcal {M}_{\varepsilon}\} has a finite open cover of balls with radius \frac 12\delta in L^2(\mathbb{R}^n) . Since \delta > 0 is arbitrary, we obtain that the set \{\zeta(s):\zeta\in\mathcal {M}_{\varepsilon}\} is percompact in H . Then from (5.9) and (5.11), we obtain that \mathcal{M}_{\varepsilon} is equicontinuous in C([-\rho, 0], H) . Therefore, by the Ascoli-Arzel \grave{a} theorem we deduce that \mathcal {M}_{\varepsilon} is precompact in C([-\rho, 0], H) , which along with (5.13) shows that \{m_k\}_{k = 1}^\infty is tight on C([-\rho, 0], H) .
Step 2. We prove the existence of invariant measures of problems (1.1) and (1.2). Since the sequence \{\mathscr{M}_k\}_{k = 1}^\infty is tight on C([-\rho, 0];H) , there exists a probability measure m on C([-\rho, 0];H) , we take a subsequence of \{\mathscr{M}_k\} (not rebel) such that \mathscr{M}_k\rightarrow m, \ \ as\ \ k\rightarrow \infty. In the following, we prove \mathscr{M} is an invariant measure of (1.1) and (1.2). Applying (5.1) and the Chapman-Kolmogorov equation, we obtain that for every t\geq0 and every \phi:C([-\rho, 0];H)\rightarrow \mathbb{R} ,
\begin{align*} &\ \ \ \ \int_{C([-\rho,0];H)}\phi(v)d\mathscr{M}(v) = \lim\limits_{k\rightarrow \infty}\frac1k\int^{k+\rho}_\rho\left(\int_{C([-\rho,0];H)}\phi(v)p(0,0;s,dv)\right)ds\\& = \lim\limits_{k\rightarrow \infty}\frac1k\int^{k+\rho-t}_{\rho-t}\left(\int_{C([-\rho,0];H)}\phi(v)p(0,0;s+t,dv)\right)ds\\& = \lim\limits_{k\rightarrow \infty}\frac1k\int^{k+\rho}_\rho\left(\int_{C([-\rho,0];H)}\phi(v)p(0,0;s+t,dv)\right)ds\\& = \lim\limits_{k\rightarrow \infty}\frac1k\int^{k+\rho}_\rho\left(\int_{C([-\rho,0];H)}\left(\int_{C([-\rho,0];H)}\phi(v)p(s,\varphi;s+t,dv)\right)p(0,0;s,d\varphi)\right)ds\\& = \lim\limits_{k\rightarrow \infty}\frac1k\int^{k+\rho}_\rho\left(\int_{C([-\rho,0];H)}\left(\int_{C([-\rho,0];H)}\phi(v)p(0,\varphi;t,dv)\right)p(0,0;s,d\varphi)\right)ds\\& = \int_{C([-\rho,0];H)}\left(\int_{C([-\rho,0];H)}\phi(v)p(0,\varphi;t,dv)\right)d\mathscr{M}(\varphi)\\& = \int_{C([-\rho,0];H)}(p_{0,t}\phi)(\varphi)d\mathscr{M}(\varphi), \end{align*} |
which completes the proof.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
The authors are grateful to the anonymous referees whose suggestions have in our opinion, greatly improved the paper. This work is partially supported by the NSF of Shandong Province (No. ZR 2021MA055) and USA Simons Foundation (No. 628308).
The authors declare there is no conflict of interest.
[1] | A Aiman (2020) Enough rice to last up to 6 months, says Khazanah researcher. FMT News. Available from: https://www.freemalaysiatoday.com/category/nation/2020/04/03/enough-rice-to-last-up-to-6-months-says-khazanah-researcher/ |
[2] | Bernama (2021) Malaysia to raise rice buffer stock to 290,000 metric tonnes by 2023. Malay Mail. Available from: https://www.malaymail.com/news/malaysia/2021/09/04/malaysia-to-raise-rice-buffer-stock-to-290000-metric-tonnes-by-2023/2002982 |
[3] | Department of Statistics Malaysia (2021) Selected agricultural indicators, Malaysia, 2021. Available from: https://www.dosm.gov.my/v1/index.php?r=column/pdfPrev&id=TDV1YU4yc1Z0dUVyZ0xPV0ptRlhWQT09 |
[4] | Che Omar S, Shaharudin A, Tumin SA (2019) The status of the paddy and rice industry in Malaysia. Khazanah Research Institute. Available from: http://www.krinstitute.org/assets/contentMS/img/template/editor/20190409_RiceReport_Full Report_Final.pdf |
[5] | Ministry of Agriculture and Food Industries (2021) MAFI mengalu-alukan pembangunan benih padi baharu IS21 oleh Agensi Nuklear Malaysia, MOSTI. Available from: https://www.mafi.gov.my/documents/20182/269754/SIARAN+MEDIA+MAFI+VARIETI+BENIH+PADI+%2820+NOVEMBER+2021%29-min.pdf/48681d83-e1c5-425f-9901-6dae0a537b7d |
[6] | Malaysian Agriculture Research and Development Institute (2021) Varieti padi MARDI Seri Waja dan Kembangsari serta Teknologi kadar boleh ubah bagi tanaman padi bantu petani tingkat hasil padi negara (Press release). Available from: https://www.mafi.gov.my/documents/20182/269754/Siaran+Media+-+Varieti+Padi+MARDI+MR+315+dan+MRQ+104+serta+%E2%80%98Teknologi+Kadar+Boleh+Ubah+Bagi+Tanaman+Padi%E2%80%99+Bantu+Petani+Tingkat+Hasil+Padi+Negara.pdf/5ac21320-cfde-455c-ad8d-ad7a3e438e9e |
[7] |
Ratjens S, Mortensen S, Kumpf A, et al. (2018) Embryogenic callus as target for efficient transformation of Cyclamen persicum enabling gene function studies. Front Plant Sci 9: 1035. https://doi.org/10.3389/fpls.2018.01035 doi: 10.3389/fpls.2018.01035
![]() |
[8] |
Nguyen THN, Winkelmann T, Debener T (2020) Genetic analysis of callus formation in a diversity panel of 96 rose genotypes. Plant Cell Tiss Organ Cult 142: 505-517. https://doi.org/10.1007/s11240-020-01875-6 doi: 10.1007/s11240-020-01875-6
![]() |
[9] |
Li CJ, Li XJ, Lin XQ, et al. (2021) Genotypic variation in the response to embryogenic callus induction and regeneration in Saccharum spontaneum. Plant Genet Resour 19: 153-158. https://doi.org/10.1017/S1479262121000198 doi: 10.1017/S1479262121000198
![]() |
[10] |
Slimani C, El Goumi Y, Rais C, et al. (2022) Optimization of callogenesis/caulogenesis induction protocol in saffron plant (Crocus sativus L.) using response surface methodology. Biointerface Res Appl Chem 12: 4731-4746. https://doi.org/10.33263/BRIAC124.47314746 doi: 10.33263/BRIAC124.47314746
![]() |
[11] |
Hameg R, Arteta TA, Landin M, et al. (2020) Modeling and optimizing culture medium mineral composition for in vitro propagation of Actinidia arguta. Front Plant Sci 11: 554905. https://doi.org/10.3389/fpls.2020.554905 doi: 10.3389/fpls.2020.554905
![]() |
[12] |
Hesami M, Jones AMP (2021) Modeling and optimizing callus growth and development in Cannabis sativa using random forest and support vector machine in combination with a genetic algorithm. Appl Microbiol Biotechnol 105: 5201-5212. https://doi.org/10.1007/s00253-021-11375-y doi: 10.1007/s00253-021-11375-y
![]() |
[13] |
Murashige T, Skoog F (1962) A revised medium for rapid growth and bioassays with tobacco tissue cultures. Physiol Plantarum 15: 473-497. https://doi.org/10.1111/j.1399-3054.1962.tb08052.x doi: 10.1111/j.1399-3054.1962.tb08052.x
![]() |
[14] |
Niedz RP, Evens TJ (2007) Regulating plant tissue growth by mineral nutrition. In Vitro Cell Dev Biol-Plant 43: 370-381. https://doi.org/10.1007/s11627-007-9062-5 doi: 10.1007/s11627-007-9062-5
![]() |
[15] |
Gao F, Peng CX, Wang H, et al. (2021). Selection of culture conditions for callus induction and proliferation by somatic embryogenesis of Pinus koraiensis. J For Res 32: 483-491. https://doi.org/10.1007/s11676-020-01147-1 doi: 10.1007/s11676-020-01147-1
![]() |
[16] | Skoog F, Miller CO (1957) Chemical regulation of growth and organ formation in plant tissues cultured in vitro. Symp Soc Exp Biol 11: 118-130. |
[17] |
Ikeuchi M, Sugimoto K, Iwase A (2013) Plant callus: Mechanisms of induction and repression. Plant Cell 25: 3159-3173. https://doi.org/10.1105/tpc.113.116053 doi: 10.1105/tpc.113.116053
![]() |
[18] |
Garcia C, de Almeida AF, Costa M, et al. (2019) Abnormalities in somatic embryogenesis caused by 2, 4-D: An overview. Plant Cell Tiss Organ Cult 137: 193-212. https://doi.org/10.1007/s11240-019-01569-8 doi: 10.1007/s11240-019-01569-8
![]() |
[19] |
Méndez-Hernández HA, Ledezma-Rodríguez M, Avilez-Montalvo RN, et al. (2019) Signaling overview of plant somatic embryogenesis. Front Plant Sci 10: 77. https://doi.org/10.3389/fpls.2019.00077 doi: 10.3389/fpls.2019.00077
![]() |
[20] |
Barman HN, Hoque ME, Roy RK, et al. (2016) Mature embryo-based in vitro regeneration of indica rice cultivars for high frequency plantlets production. Bangladesh Rice J 20: 81-87. https://doi.org/10.3329/brj.v20i2.34132 doi: 10.3329/brj.v20i2.34132
![]() |
[21] |
Mostafiz SB, Wagiran A (2018) Efficient callus induction and regeneration in selected indica rice. Agronomy 8: 77. https://doi.org/10.3390/agronomy8050077 doi: 10.3390/agronomy8050077
![]() |
[22] | Singh S, Kumar A, Rana V, et al. (2018) In Vitro callus induction and plant regeneration in basmati rice (Oryza sativa L.) varieties. J Pharmacogn Phytochem 7: 65-69. |
[23] |
Trunjaruen A, Raso S, Maneerattanarungroj P, et al. (2020) Effects of cultivation media on in vitro callus induction and regeneration capabilities of pakaumpuel rice (Oryza sativa L.), Thai rice landrace. Walailak J Sci Tech 17: 37-46. https://doi.org/10.48048/wjst.2020.5349 doi: 10.48048/wjst.2020.5349
![]() |
[24] |
Feher A, Pasternak TP, Dudits D (2003) Transition of somatic plant cells to an embryogenic state. Plant Cell Tiss Organ Cult 74: 201-228. https://doi.org/10.1023/A:1024033216561 doi: 10.1023/A:1024033216561
![]() |
[25] |
Zavattieri MA, Frederico AM, Lima M, et al. (2010) Induction of somatic embryogenesis as an example of stress-related plant reactions. Electron J Biotechn 13: 1-9. https://doi.org/10.2225/vol13-issue1-fulltext-4 doi: 10.2225/vol13-issue1-fulltext-4
![]() |
[26] |
Pěnčík A, Turečková V, PaulišicPaulišic, et al. (2015) Ammonium regulates embryogenic potential in Cucurbita pepo through pH-mediated changes in endogenous auxin and abscisic acid. Plant Cell Tiss Organ Cult 122: 89-100. https://doi.org/10.1007/s11240-015-0752-0 doi: 10.1007/s11240-015-0752-0
![]() |
[27] | Loyola-Vargas VM, Ochoa-Alejo N (2016) Somatic embryogenesis. An overview. In: Somatic embryogenesis: Fundamental aspects and applications. Springer, Cham. https://doi.org/10.1007/978-3-319-33705-0_1 |
[28] |
Rizwan HM, Irshad M, He BZ, et al. (2020) Role of reduced nitrogen for induction of embryogenic callus induction and regeneration of plantlets in Abelmoschus esculentus L. S Afr J Bot 130: 300-307. https://doi.org/10.1016/j.sajb.2020.01.016 doi: 10.1016/j.sajb.2020.01.016
![]() |
[29] |
Pasternak TP, Prinsen E, Ayaydin F, et al. (2002) The role of auxin, pH, and stress in the activation of embryogenic cell division in leaf protoplast-derived cells of alfalfa. Plant Physiol 129: 1807-1819. https://doi.org/10.1104/pp.000810 doi: 10.1104/pp.000810
![]() |
[30] |
Vondrakova Z, Eliasova K, Fischerova L, et al. (2011) The role of auxins in somatic embryogenesis of Abies alba. Cent Eur J Biol 6: 587-596. https://doi.org/10.2478/s11535-011-0035-7 doi: 10.2478/s11535-011-0035-7
![]() |
[31] | Nic-Can GI, Loyola-Vargas VM (2016) The role of the auxins during somatic embryogenesis. In: Somatic embryogenesis: Fundamental aspects and applications. Springer, Cham. https://doi.org/10.1007/978-3-319-33705-0_10 |
[32] |
Michalczuk L, Ribnicky DM, Cooke TJ, et al. (1992) Regulation of indole-3-acetic acid biosynthetic pathways in carrot cell cultures. Plant Physiol 100: 1346-1353. https://doi.org/10.1104/pp.100.3.1346 doi: 10.1104/pp.100.3.1346
![]() |
[33] | Mostafiz S (2019) Analysis of embryogenic callus induction and regeneration of indica rice variety of Malaysia. PhD Thesis, Universiti Teknologi Malaysia. Available from: http://eprints.utm.my/id/eprint/81396 |
[34] |
Kalhori N, Nulit R, Go R, et al. (2017) Selection, characterizations and somatic embryogenesis of Malaysian salt-tolerant rice (Oryza sativa cv. MR219) through callogenesis. Int J Agric Biol 19: 157-163. https://doi.org/10.17957/IJAB/15.0258 doi: 10.17957/IJAB/15.0258
![]() |
[35] |
Dalila ZD, Jaafar H, Manaf AA (2013) Effects of 2, 4-D and kinetin on callus induction of Barringtonia racemosa leaf and endosperm explants in different types of basal media. Asian J Plant Sci 12: 21-27. https://doi.org/10.3923/ajps.2013.21.27 doi: 10.3923/ajps.2013.21.27
![]() |
[36] | Karimian R, Lahouti M, Davarpanah SJ (2014) Effects of different concentrations of 2, 4-D and kinetin on callogenesis of Taxus revifolia Nutt. J Appl Biotechnol Rep 1: 167-170. |
[37] |
Konar S, Karmakar J, Ray A, et al. (2018) Regeneration of plantlets through somatic embryogenesis from root derived calli of Hibiscus sabdariffa L. (roselle) and assessment of genetic stability by flow cytometry and ISSR analysis. PLoS One 13: e0202324. https://doi.org/10.1371/journal.pone.0202324 doi: 10.1371/journal.pone.0202324
![]() |
[38] |
Shekar S, Sankepally R, Singh B (2016) Optimization of regeneration using differential growth regulators in indica rice cultivars. 3 Biotech 6: 19. https://doi.org/10.1007/s13205-015-0343-0 doi: 10.1007/s13205-015-0343-0
![]() |
[39] | Verma D, Joshi R, Shukla A, et al. (2011) Protocol for in vitro somatic embryogenesis and regeneration of rice (Oryza sativa L.). Indian J Exp Biol 49: 958-963. |
[40] | Le Bris M (2017). Hormones in growth and development. In: Reference module in life sciences. Elsevier. 364-369. https://doi.org/10.1016/B978-0-12-809633-8.05058-5 |
[41] |
Sagare AP, Lee YL, Lin TC, et al. (2000) Cytokinin-induced somatic embryogenesis and plant regeneration in Corydalis yanhusuo (Fumariaceae)-A medicinal plant. Plant Sci 160: 139-147. https://doi.org/10.1016/S0168-9452(00)00377-0 doi: 10.1016/S0168-9452(00)00377-0
![]() |
[42] |
Naranjo EJ, Betin OF, Trujillo AIU, et al. (2016) Effect of genotype on the in vitro regeneration of Stevia rebaudiana via somatic embryogenesis. Acta Biol Colomb 21: 87-98. https://doi.org/10.15446/abc.v21n1.47382 doi: 10.15446/abc.v21n1.47382
![]() |
[43] | Mangena P (2018) The role of plant genotype, culture medium and Agrobacterium on soybean plantlets regeneration during genetic transformation. In: Transgenic crops: Emerging trends and future perspectives. https://doi.org/10.5772/intechopen.78773 |
[44] |
Sabbadini S, Capriotti L, Molesini B, et al. (2019) Comparison of regeneration capacity and Agrobacterium-mediated cell transformation efficiency of different cultivars and rootstocks of Vitis spp. via organogenesis. Sci Rep 9: 582. https://doi.org/10.1038/s41598-018-37335-7 doi: 10.1038/s41598-018-37335-7
![]() |
[45] |
Konôpková J, Košútová D, Ferus P (2020) Genotype-specific requirements for in vitro culture initiation and multiplication of Magnolia taxa. Folia Oecologica 47: 34-44. https://doi.org/10.2478/foecol-2020-0005 doi: 10.2478/foecol-2020-0005
![]() |
[46] |
Lin YJ, Zhang QF (2005) Optimising the tissue culture conditions for high efficiency transformation of indica rice. Plant Cell Rep 23: 540-547. https://doi.org/10.1007/s00299-004-0843-6 doi: 10.1007/s00299-004-0843-6
![]() |
[47] | Rahim H, Amin EEEA, Mat MZ (2021) Penilaian tahap penggunaan teknologi padi MARDI di Muda Agricultural Development Authority (MADA). Available from: http://ebuletin.mardi.gov.my/buletin/23/Hairazi.pdf |
[48] | Bernama (2021) MARDI henti jual benih padi MR 219. Berita Harian. Available from: https://www.bharian.com.my/berita/nasional/2021/10/876638/mardi-henti-jual-benih-padi-mr-219 |
[49] |
Karthikeyan A, Pandian STK, Ramesh M (2009) High frequency plant regeneration from embryogenic callus of a popular indica rice (Oryza sativa L.). Physiol Mol Biol Plants 15: 371-375. https://doi.org/10.1007/s12298-009-0042-6 doi: 10.1007/s12298-009-0042-6
![]() |
[50] | Sah SK, Kaur A, Sandhu JS (2014) High frequency embryogenic callus induction and whole plant regeneration in japonica rice cv. Kitaake. J Rice Res 2: 125. https://doi.org/0.4172/jrr.1000125 |
[51] |
Abiri R, Maziah M, Shaharuddin NA, et al. (2017) Enhancing somatic embryogenesis of Malaysian rice cultivar MR219 using adjuvant materials in a high-efficiency protocol. Int J Environ Sci Technol 14: 1091-1108. https://doi.org/10.1007/s13762-016-1221-y doi: 10.1007/s13762-016-1221-y
![]() |
[52] |
Sahoo KK, Tripathi AK, Pareek A, et al. (2011) An improved protocol for efficient transformation and regeneration of diverse indica rice cultivars. Plant Methods 7: 49. https://doi.org/10.1186/1746-4811-7-49 doi: 10.1186/1746-4811-7-49
![]() |
[53] |
Carsono N, Yoshida T (2006) Identification of callus induction potential of 15 Indonesian rice genotypes. Plant Prod Sci 9: 65-70. https://doi.org/10.1626/pps.9.65 doi: 10.1626/pps.9.65
![]() |
[54] | Bhatia S (2015) Plant tissue culture. In: Modern applications of plant biotechnology in pharmaceutical sciences. Academic Press. https://doi.org/10.1016/C2014-0-02123-5 |
[55] |
Mastuti R, Munawarti A, Firdiana ER (2017) The combination effect of auxin and cytokinin on in vitro callus formation of Physalis angulata L.-A medicinal plant. AIP Conf Proc 1908: 040007. https://doi.org/10.1063/1.5012721 doi: 10.1063/1.5012721
![]() |
[56] |
Vennapusa AR, Vemanna RS, Reddy BHR, et al. (2015) An efficient callus induction and regeneration protocol for a drought tolerant rice indica genotype AC39020. J Plant Sci 3: 248-254. https://doi.org/10.11648/j.jps.20150305.11 doi: 10.11648/j.jps.20150305.11
![]() |
[57] |
Ramakrishna D, Shasthree T (2016) High efficient somatic embryogenesis development from leaf cultures of Citrullus colocynthis (L.) Schrad for generating true type clones. Physiol Mol Biol Plants 22: 279-285. https://doi.org/10.1007/s12298-016-0357-z doi: 10.1007/s12298-016-0357-z
![]() |
[58] |
Krishnan SRS, Siril EA (2017) Auxin and nutritional stress coupled somatic embryogenesis in Oldenlandia umbellata L. Physiol Mol Biol Plants 23: 471-475. https://doi.org/10.1007/s12298-017-0425-z doi: 10.1007/s12298-017-0425-z
![]() |
[59] |
Visarada KBRS, Sailaja M, Sarma NP (2002) Effect of callus induction media on morphology of embryogenic calli in rice genotypes. Biologia Plantarum 45: 495-502. https://doi.org/10.1023/A:1022323221513 doi: 10.1023/A:1022323221513
![]() |
[60] |
Gaj MD (2004) Factors influencing somatic embryogenesis induction and plant regeneration with particular reference to Arabidopsis thaliana (L.) Heynh. Plant Growth Regulation 43: 27-47. https://doi.org/10.1023/B:GROW.0000038275.29262.fb doi: 10.1023/B:GROW.0000038275.29262.fb
![]() |
[61] |
Mohd Din ARJ, Ahmad FI, Wagiran A, et al. (2016) Improvement of efficient in vitro regeneration potential of mature callus induced from Malaysian upland rice seed (Oryza sativa cv. Panderas). Saudi J Biol Sci 23: S69-S77. https://doi.org/10.1016/j.sjbs.2015.10.022 doi: 10.1016/j.sjbs.2015.10.022
![]() |
[62] |
Rizwan HM, Irshad M, He BZ, et al. (2018) Silver nitrate (AgNO3) boosted high-frequency multiple shoot regeneration from cotyledonary node explants of okra (Abelmoschus esculentus L.). Appl Ecol Env Res 16: 3421-3435. http://doi.org/10.15666/aeer/1603_34213435 doi: 10.15666/aeer/1603_34213435
![]() |
[63] |
Rizwan HM, Yang Q, Yousef AF, et al. (2021) Establishment of a novel and efficient Agrobacterium-mediated in planta transformation system for passion fruit (Passiflora edulis). Plants 10: 2549. https://doi.org/10.3390/plants10112459 doi: 10.3390/plants10112459
![]() |
[64] |
Poeaim A, Poeaim S, Poraha R, et al. (2016) Optimization for callus induction and plant regeneration from mature seeds of Thai rice variety: Nam Roo (Oryza sativa L.). Bioeng Biosci 4: 95-99. https://doi.org/10.13189/bb.2016.040504 doi: 10.13189/bb.2016.040504
![]() |
[65] |
Wani SH, Sanghera GS, Gosal SS (2011) An efficient and reproducible method for regeneration of whole plants from mature seeds of a high yielding indica rice (Oryza sativa L.) variety PAU 201. New Biotechnol 28: 418-422. https://doi.org/10.1016/j.nbt.2011.02.006 doi: 10.1016/j.nbt.2011.02.006
![]() |
[66] |
Colomba EL, Grunberg K, Griffa S, et al. (2006) The effect of genotype and culture medium on somatic embryogenesis and plant regeneration from mature embryos of fourteen apomictic cultivars of buffel grass (Cenchrus ciliaris L.). Grass Forage Sci 61: 2-8. https://doi.org/10.1111/j.1365-2494.2006.00499.x doi: 10.1111/j.1365-2494.2006.00499.x
![]() |
[67] |
Hofmann N (2014) Getting to the root of regeneration: Adventitious rooting and callus formation. Plant Cell 26: 845. https://doi.org/10.1105/tpc.114.125096 doi: 10.1105/tpc.114.125096
![]() |
[68] |
Yu J, Liu W, Liu J, et al. (2017) Auxin control of root organogenesis from callus in tissue culture. Front Plant Sci 8: 1385. https://doi.org/10.3389/fpls.2017.01385 doi: 10.3389/fpls.2017.01385
![]() |
[69] |
Hiei Y, Ohta S, Komari T, et al. (1994) Efficient transformation of rice (Oryza sativa L.) mediated by Agrobacterium and sequence analysis of the boundaries of the T-DNA. Plant J 6: 271-282. https://doi.org/10.1046/j.1365-313x.1994.6020271.x doi: 10.1046/j.1365-313x.1994.6020271.x
![]() |
[70] |
Al-Forkan M, Rahim MA, Chowdhury T, et al. (2005). Development of highly in vitro callogenesis and regeneration system for some salt tolerant rice (Oryza sativa L.) cultivars of Bangladesh. Biotechnology 4: 230-234. https://doi.org/10.3923/biotech.2005.230.234 doi: 10.3923/biotech.2005.230.234
![]() |
[71] |
Sahoo KK, Tripathi AK, Pareek A, et al. (2011) An improved protocol for efficient transformation and regeneration of diverse indica rice cultivars. Plant Methods 7: 49. https://doi.org/10.1186/1746-4811-7-49 doi: 10.1186/1746-4811-7-49
![]() |
[72] | Jimenez VM, Thomas C (2005) Participation of plant hormones in determination and progression of somatic embryogenesis. In: Somatic embryogenesis. Berlin, Heidelberg: Springer, 2: 103-118. https://doi.org/10.1007/7089_034 |
[73] |
Jimenez VM, Bangerth F (2001) Endogenous hormone levels in explants and in embryogenic and non-embryogenic cultures of carrot. Physiol Plant 111: 389-395. https://doi.org/10.1034/j.1399-3054.2001.1110317.x doi: 10.1034/j.1399-3054.2001.1110317.x
![]() |
[74] |
Mariani TS, Miyake H, Takeoka Y (1998) Changes in surface structure during direct somatic embryogenesis in rice scutellum observed by scanning electron microscopy. Plant Prod Sci 1: 223-231. https://doi.org/10.1626/pps.1.223 doi: 10.1626/pps.1.223
![]() |
[75] | Poraha R, Poeaim A, Pongjaroenkit S (2016) Callus induction and plant regeneration on optimization of the culture conditions in Jow Haw rice (Oryza sativa L.). J Agric Technol 12: 241-248. |
[76] | Remme RN, Snigdha SS, Islam MM, et al. (2017). In vitro morphogenesis in two indigenous rice (Oryza sativa L.) cultivars through dehusked seed culture. Khulna Univ Stud 14: 15-26. |
[77] |
Roly ZY, Islam MM, Shaekh MPE, et al. (2014) In vitro callus induction and regeneration potentiality of aromatic rice (Oryza sativa L.) cultivars in differential growth regulators. Int J Appl Sci Biotechnol 2: 160-167. https://doi.org/10.3126/ijasbt.v2i2.10313 doi: 10.3126/ijasbt.v2i2.10313
![]() |
[78] | Meneses A, Flores D, Muñoz M, et al. (2005) Effect of 2, 4-D, hydric stress and light on indica rice (Oryza sativa) somatic embryogenesis. Rev Biol Trop 53: 361-368. |
[79] |
Mostafiz SB, Wagiran A, Johan NS, et al. (2018) The effects of temperature on callus induction and regeneration in selected Malaysian rice cultivar indica. Sains Malaysiana 47: 2647-2655. https://doi.org/10.17576/jsm-2018-4711-07 doi: 10.17576/jsm-2018-4711-07
![]() |
[80] | Zaidi MA, Narayanan M, Sardana R, et al. (2006) Optimizing tissue culture media for efficient transformation of different indica rice genotypes. Agron Res 4: 563-575. |
[81] |
Pérez-Bernal M, Rigo MD, Díaz CAH, et al. (2009) Callus induction and plant regeneration of two Cuban rice cultivars using different seed explants and amino acid supplements. Ann Trop Res 15: 1-15. https://doi.org/10.32945/atr3121.2009 doi: 10.32945/atr3121.2009
![]() |
[82] |
Vennapusa AR, Vemanna RS, Reddy BHR, et al. (2015) An efficient callus induction and regeneration protocol for a drought tolerant rice indica genotype AC39020. J Plant Sci 3: 248-254. https://doi.org/10.11648/j.jps.20150305.11 doi: 10.11648/j.jps.20150305.11
![]() |
[83] |
Barman HN, Hoque ME, Roy RK, et al. (2017) Mature embryo-based in vitro regeneration of indica rice cultivars for high frequency plantlets production. Bangladesh Rice J 20: 81-87. https://doi.org/10.3329/brj.v20i2.34132 doi: 10.3329/brj.v20i2.34132
![]() |
[84] |
Dhamangaonkar SN, Shukla L (2013) Identification of callus induction potential of nine different genotypes of indica rice. J Environ Nanotechnol 2: 45-52. https://doi.org/10.13074/jent.2013.06.132013 doi: 10.13074/jent.2013.06.132013
![]() |
1. | Fabio Di Pietrantonio, Domenico Cannatà, Massimiliano Benetti, 2019, 9780128144015, 181, 10.1016/B978-0-12-814401-5.00008-6 | |
2. | Christina G. Siontorou, 2020, Chapter 25-2, 978-3-319-47405-2, 1, 10.1007/978-3-319-47405-2_25-2 | |
3. | Christina G. Siontorou, Georgia-Paraskevi Nikoleli, Marianna-Thalia Nikolelis, Dimitrios P. Nikolelis, 2019, 9780128157435, 375, 10.1016/B978-0-12-815743-5.00015-9 | |
4. | Christina G. Siontorou, 2019, Chapter 25-1, 978-3-319-47405-2, 1, 10.1007/978-3-319-47405-2_25-1 | |
5. | Christina G. Siontorou, 2022, Chapter 25, 978-3-030-23216-0, 707, 10.1007/978-3-030-23217-7_25 |