This paper presents an initial investigation into the dynamic properties of a well-known iterative method for solving nonlinear equations: Schröder's method. We characterize the degree of the rational map induced by applying the method to polynomial equations, along with other dynamical features such as the nature of extraneous fixed points and the presence of attracting cycles. Particular attention is given to the significant dynamical differences between Schröder's method and other iterative methods, notably Newton's method, with a focus on the behavior at infinity.
Citation: Víctor Galilea, José Manuel Gutiérrez. Schröder's method and the infinity point[J]. AIMS Mathematics, 2025, 10(6): 12919-12934. doi: 10.3934/math.2025581
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This paper presents an initial investigation into the dynamic properties of a well-known iterative method for solving nonlinear equations: Schröder's method. We characterize the degree of the rational map induced by applying the method to polynomial equations, along with other dynamical features such as the nature of extraneous fixed points and the presence of attracting cycles. Particular attention is given to the significant dynamical differences between Schröder's method and other iterative methods, notably Newton's method, with a focus on the behavior at infinity.
G. Caginalp proposed in [3] and [4] two phase-field system, namely,
∂u∂t−Δu+f(u)=T, | (1.1) |
∂T∂t−ΔT=−∂u∂t, | (1.2) |
called nonconserved system, and
∂u∂t+Δ2u−Δf(u)=−ΔT, | (1.3) |
∂T∂t−ΔT=−∂u∂t, | (1.4) |
called concerved system (in the sense that, when endowed with Neumann boundary conditions, the spacial average of u is conserved). In this context, u is the order parameter, T is the relative temperature (defined as T=˜T−TE, where ˜T is the absolute temperature and TE is the equilibrium melting temperature) and f is the derivative of a double-well potential F (a typical choice is F(s)=14(s2−1)2, hence the usual cubic nonlinear term f(s)=s3−s). Furthermore, we have set all physical parameters equal to one. These systems have been introduced to model phase transition phenomena, such as melting-solidication phenomena, and have been much studied from a mathematical point of view. We refer the reader to, e.g., [3,4,5,8,9,10,12,13,14,15,16,18,19,21,22,23,25].
Both systems are based on the (total Ginzburg-Landau) free energy
ΨGL=∫Ω(12|∇u|2+F(u)−uT−12T2)dx, | (1.5) |
where Ω is the domain occupied by the system (we assume here that it is a bounded and regular domain of R3, with boundary Γ), and the enthalpy
H=u+T. | (1.6) |
As far as the evolution equations for the order parameter are concerned, one postulates the relaxation dynamics (with relaxation parameter set equal to one)
∂u∂u=−DΨGLDu, | (1.7) |
for the nonconserved model, and
∂u∂u=ΔDΨGLDu, | (1.8) |
for the conserved one, where DDu denotes a variational derivative with respect to u, which yields (1.1) and (1.3), respectively. Then, we have the energy equation
∂H∂t=−divq, | (1.9) |
where q is the heat flux. Assuming finally the usual Fourier law for heat conduction,
q=−∇T, | (1.10) |
we obtain (1.2).
In (1.5), the term |∇u|2 models short-ranged interactions. It is however interesting to note that such a term is obtained by truncation of higher-order ones; it can also be seen as a first-order approximation of a nonlocal term accounting for long-ranged interactions [11].
G. Caginalp and Esenturk recently proposed in [6] (see also [20]) higher-order phase-field models in order to account for anisotropic interfaces (see also [7] for other approaches which, however, do not provide an explicit way to compute the anisotropy). More precisely, these autors proposed the following modified (total) free energy
ΨHOGL=∫Ω(12∑ki=1∑|β|=iaβ|Dβu|2+F(u)−uT−12T2)dx,k∈N, | (1.11) |
where, for β=(k1,k2,k3)∈(N∪{0})3,
|β|=k1+k2+k3 |
and, for β≠(0,0,0),
Dβ=∂|β|∂xk11∂xk22∂xk33 |
(we agree that D(0,0,0)v=v).
A. Miranville studied in [17] the corresponding nonconserved higher-order phase-field system.
As far as the conserved case is concerned, the above generalized free energy yields, procceding as above, the following evolution equation for the order parameter u:
∂u∂t−Δ∑ki=1(−1)i∑|β|=iaβD2βu−Δf(u)=−Δ(∂α∂t−Δ∂α∂t), | (1.12) |
In particular, for k = 1 (anisotropic conserved Caginalp phase-field), we have an equation of the form
∂u∂t+Δ∑3i=1ai∂2u∂x2i−Δf(u)=−Δ(∂α∂t−Δ∂α∂t) |
and, for k = 2 (fourth-order anisotropic conserved Caginalp phase-field system), we have an equation of the form
∂u∂t−Δ∑3i,j=1aij∂4u∂x2i∂x2j+Δ∑3i=1bi∂2u∂x2i−Δf(u)=−Δ(∂α∂t−Δ∂α∂t). |
L. Cherfils A. Miranville and S. Peng have studied in [8] the corresponding higher-order isotropic equation (without the coupling with the temperature), namely, the equation
∂u∂t−ΔP(−Δ)u−Δf(u)=0, |
where
P(s)=∑ki=1aisi,ak>0,k⩾1, |
endowed with the Dirichlet/Navier boundary conditions
u=Δu=...=Δku=0onΓ. |
Our aim in this paper is to study the model consisting of the higher-order anisotropic equation (1.12) and the temperature equation
∂2α∂t2−Δ∂2α∂t2−Δ∂α∂t−Δα=−∂u∂t. | (1.13) |
In particular, we obtain the existence and uniqueness of solutions.
We consider the following initial and boundary value problem, for k∈N, k⩾2 (the case k = 1 can be treated as in the original conserved system; see, e.g., [23]):
∂u∂t−Δ∑ki=1(−1)i∑|β|=iaβD2βu−Δf(u)=−Δ(∂α∂t−Δ∂α∂t), | (2.1) |
∂2α∂t2−Δ∂2α∂t2−Δ∂α∂t−Δα=−∂u∂t, | (2.2) |
Dβu=α=0onΓ,|β|⩽k, | (2.3) |
u|t=0=u0,α|t=0=α0,∂α∂t|t=0=α1. | (2.4) |
We assume that
aβ>0,|β|=k, | (2.5) |
and we introduce the elliptic operator Ak defined by
⟨Akv,w⟩H−k(Ω),Hk0(Ω)=∑|β|=kaβ((Dβv,Dβw)), | (2.6) |
where H−k(Ω) is the topological dual of Hk0(Ω). Furthermore, ((., .)) denotes the usual L2-scalar product, with associated norm ‖.‖. More generally, we denote by ‖.‖X the norm on the Banach space X; we also set ‖.‖−1=‖(−Δ)−12.‖, where (−Δ)−1 denotes the inverse minus Laplace operator associated with Dirichlet boudary conditions. We can note that
(v,w)∈Hk0(Ω)2↦∑|β|=kaβ((Dβv,Dβw)) |
is bilinear, symmetric, continuous and coercive, so that
Ak:Hk0(Ω)→H−k(Ω) |
is indeed well defined. It then follows from elliptic regularity results for linear elliptic operators of order 2k (see [1] and [2]) that Ak is a strictly positive, selfadjoint and unbounded linear operator with compact inverse, with domain
D(Ak)=H2k(Ω)∩Hk0(Ω), |
where, for v∈D(Ak),
Akv=(−1)k∑|β|=kaβD2βv. |
We further note that D(A12k)=Hk0(Ω) and, for (v,w)∈D(A12k)2,
((A12kv,A12kw))=∑|β|=kaβ((Dβv,Dβw)). |
We finally note that (see, e.g., [24]) ‖Ak.‖ (resp., ‖A12k.‖) is equivalent to the usual H2k-norm (resp., Hk-norm) on D(Ak) (resp., D(A12k)).
Similarly, we can define the linear operator ¯Ak=−ΔAk
ˉAk:Hk+10(Ω)→H−k−1(Ω) |
which is a strictly positive, selfadjoint and unbounded linear operator with compact inverse, with domain
D(ˉAk)=H2k+2(Ω)∩Hk+10(Ω), |
where, for v∈D(ˉAk),
ˉAkv=(−1)k+1Δ∑|β|=kaβD2βv. |
Furthermore, D(ˉA12k)=Hk+10(Ω) and, for (v,w)∈D(ˉA12k),
((ˉA12kv,ˉA12kw))=∑|β|=kaβ((∇Dβv,∇Dβw)). |
Besides ‖ˉAk.‖ (resp., ‖ˉA12k.‖) is equivalent to the usual H2k+2-norm (resp., Hk+1-norm) on D(ˉAk) (resp., D(ˉA12k)).
We finally consider the operator ˜Ak=(−Δ)−1Ak, where
˜Ak:Hk−10(Ω)→H−k+1(Ω); |
note that, as −Δ and Ak commute, then the same holds for (−Δ)−1 and Ak, so that ˜Ak=Ak(−Δ)−1.
We have the (see [17])
Lemme 2.1. The operator ˜Ak is a strictly positive, selfadjoint and unbounded linear operator with compact inverse, with domain
D(˜Ak)=H2k−2(Ω)∩Hk−10(Ω), |
where, for v∈D(˜Ak)
˜Akv=(−1)k∑|β|=kaβD2β(−Δ)−1v. |
Furthermore, D(˜A12k)=Hk−10(Ω) and, for (v,w)∈D(˜A12k),
((˜A12kv,˜A12kw))=∑|β|=kaβ((Dβ(−Δ)−12v,Dβ(−Δ)−12w)). |
Besides ‖˜Ak.‖ (resp., ‖˜A12k.‖) is equivalent to the usual H2k−2-norm (resp., Hk−1-norm) on D(˜Ak) (resp., D(˜A12k)).
Proof. We first note that ˜Ak clearly is linear and unbounded. Then, since (−Δ)−1 and Ak commute, it easily follows that ˜Ak is selfadjoint.
Next, the domain of ˜Ak is defined by
D(˜Ak)={v∈Hk−10(Ω),˜Akv∈L2(Ω)}. |
Noting that ˜Akv=f,f∈L2(Ω),v∈D(˜Ak), is equivalent to Akv=−Δf, where −Δf∈H2(Ω)′, it follows from the elliptic regularity results of [1] and [2] that v∈H2k−2(Ω), so that D(˜Ak)=H2k−2(Ω)∩Hk−10(Ω).
Noting then that ˜A−1k maps L2(Ω) onto H2k−2(Ω) and recalling that k⩾2, we deduce that ˜Ak has compact inverse.
We now note that, considering the spectral properties of −Δ and Ak (see, e.g., [24]) and recalling that these two operators commute, −Δ and Ak have a spectral basis formed of common eigenvectors. This yields that, ∀s1,s2∈R, (−Δ)s1 and As2k commute.
Having this, we see that ˜A12k=(−Δ)−12A12k, so that D(˜A12k)=Hk−10(Ω), and for (v,w)∈D(˜A12k)2,
((˜A12kv,˜A12kw))=∑|β|=kaβ((Dβ(−Δ)−12v,Dβ(−Δ)−12w)). |
Finally, as far as the equivalences of norms are concerned, we can note that, for instance, the norm ‖˜A12k.‖ is equivalent to the norm ‖(−Δ)−12.‖Hk(Ω) and, thus, to the norm ‖(−Δ)k−12.‖.
Having this, we rewrite (2.1) as
∂u∂t−ΔAku−ΔBku−Δf(u)=−Δ(∂α∂t−Δ∂α∂t), | (2.7) |
where
Bkv=∑k−1i=1(−1)i∑|β|=iaβD2βv. |
As far as the nonlinear term f is concerned, we assume that
f∈C2(R),f(0)=0, | (2.8) |
f′⩾−c0,c0⩾0, | (2.9) |
f(s)s⩾c1F(s)−c2⩾−c3,c1>0,c2,c3⩾0,s∈R, | (2.10) |
F(s)⩾c4s4−c5,c4>0,c5⩾0,s∈R, | (2.11) |
where F(s)=∫s0f(τ)dτ. In particular, the usual cubic nonlinear term f(s)=s3−s satisfies these assumptions.
Throughout the paper, the same letters c, c' and c" denote (generally positive) constants which may vary from line to line. Similary, the same letter Q denotes (positive) monotone increasing (with respect to each argument) and continuous functions which may vary from line to line.
We multiply (2.7) by (−Δ)−1∂u∂t and (2.2) by ∂α∂t−Δ∂α∂t, sum the two resulting equalities and integrate over Ω and by parts. This gives
ddt(‖A12ku‖2+B12k[u]+2∫ΩF(u)dx+‖∇α‖2+‖Δα‖2+‖∂α∂t−Δ∂α∂t‖2)+2‖∂u∂t‖2−1+2‖∇∂α∂t‖2+2‖Δ∂α∂t‖2=0 | (3.1) |
(note indeed that ‖∂α∂t‖2+2‖∇∂α∂t‖2+‖Δ∂α∂t‖2=‖∂α∂t−Δ∂α∂t‖2), where
B12k[u]=∑k−1i=1∑|β|=iaβ‖Dβu‖2 | (3.2) |
(note that B12k[u] is not necessarily nonnegative). We can note that, owing to the interpolation inequality
B12k[u]=∑k−1i=1∑|β|=iaβ‖Dβu‖2 | (3.3) |
‖(−Δ)i2v‖⩽c(i)‖(−Δ)m2v‖im‖v‖1−im, |
there holds
v∈Hm(Ω),i∈{1,...,m−1},m∈N,m⩾2, | (3.4) |
This yields, employing (2.11),
|B12k[u]|⩽12‖A12ku‖2+c‖u‖2. |
whence
\|A_{k}^{\frac{1}{2}}u\|^{2}+B_{k}^{\frac{1}{2}}\left[u\right]+2\int_{\Omega}F(u)\mathrm{d}x\geqslant\frac{1}{2}\|A_{k}^{\frac{1}{2}}u\|^{2}+\int_{\Omega}F(u)\mathrm{d}x+c\|u\|_{L^{4}(\Omega)}^{4}-c'\|u\|^{2}-c", | (3.5) |
nothing that, owing to Young's inequality,
\|A_{k}^{\frac{1}{2}}u\|^{2}+B_{k}^{\frac{1}{2}}\left[u\right]+2\int_{\Omega}F(u)\mathrm{d}x\geqslant c(\|u\|_{H^{k}(\Omega)}^{2}+\int_{\Omega}F(u)\mathrm{d}x)-c', \quad c > 0, | (3.6) |
We then multiply (2.7) by (-\Delta)^{-1}u and have, owing to (2.10) and the interpolation inequality (3.3),
\|u\|^{2}\leqslant\epsilon\|u\|_{L^{4}(\Omega)}^{4}+c(\epsilon), \quad\forall\epsilon > 0. |
hence, proceeding as above and employing, in particular, (2.11)
\dfrac{d}{dt}\|u\|_{-1}^{2}+c(\|u\|_{H^{k}(\Omega)}^{2}+\int_{\Omega}F(u)\mathrm{d}x)\leqslant c'(\|u\|^{2}+\|\dfrac{\partial \alpha}{\partial t}\|^{2}+\|\Delta\dfrac{\partial \alpha}{\partial t}\|^{2})+c", | (3.7) |
Summing (3.1) and \delta_1 times (3.7), where \delta_1>0 is small enough, we obtain a differential inegality of the form
\dfrac{d}{dt}\|u\|_{-1}^{2}+c(\|u\|_{H^{k}(\Omega)}^{2}+\int_{\Omega}F(u)\mathrm{d}x)\leqslant c'(\|\dfrac{\partial \alpha}{\partial t}\|^{2}+\|\Delta\dfrac{\partial \alpha}{\partial t}\|^{2})+c'', \quad c>0. | (3.8) |
where
\dfrac{d}{dt}E_1+c(\|u\|_{H^{k}(\Omega)}^{2}+\int_{\Omega}F(u)\mathrm{d}x+\|\dfrac{\partial u}{\partial t}\|_{-1}^{2}+\|\dfrac{\partial \alpha}{\partial t }\|_{H^{2}(\Omega)}^{2})\leqslant c', \quad c>0, |
satisfies, owing to (3.5)
E_1=\|A_{k}^{\frac{1}{2}}u\|^{2}+B_{k}^{\frac{1}{2}}\left[u\right]+2\int_{\Omega}F(u)\mathrm{d}x+\|\nabla \alpha\|^{2}+\|\Delta \alpha\|^{2}+\|\dfrac{\partial \alpha}{\partial t}-\Delta\dfrac{\partial \alpha}{\partial t}\|^{2}+\delta_1\|u\|_{-1}^{2} | (3.9) |
Multiplying (2.2) by -\Delta\alpha, we then obtain
E_{1}\geqslant c(\|u\|_{H^{k}(\Omega)}^{2}+\int_{\Omega}F(u)\mathrm{d}x+\|\alpha\|_{H^{2}(\Omega)}^{2}+\|\dfrac{\partial \alpha}{\partial t}\|_{H^{2}(\Omega)}^{2})-c', \quad c > 0. |
which yields, employing the interpolation inequality
\dfrac{d}{dt}(\|\Delta\alpha\|^{2}-2((\dfrac{\partial\alpha} {\partial t}, \Delta\alpha))+2((\Delta\dfrac{\partial\alpha} {\partial t}, \Delta\alpha)))+\|\Delta\alpha\|^{2}\leqslant \|\dfrac{\partial u}{\partial t}\|^{2}+\|\nabla\dfrac{\partial\alpha}{\partial t}\|^{2}+\|\Delta\dfrac{\partial\alpha}{\partial t}\|^{2}, | (3.10) |
the differential inequality, with 0 < \epsilon < < 1 is small enough
\|v\|^{2}\leqslant c\|v\|_{-1}\|v\|_{H^{1}(\Omega)}, \quad v\in H_{0}^{1}(\Omega), | (3.11) |
We now differentiate (2.7) with respect to time to find, owing to (2.2),
\begin{align} & \frac{d}{dt}(\|\Delta \alpha {{\|}^{2}}-2((\frac{\partial \alpha }{\partial t}, \Delta \alpha ))+2((\Delta \frac{\partial \alpha }{\partial t}, \Delta \alpha )))+c\|\alpha \|_{{{H}^{2}}(\Omega )}^{2} \\ & \le {c}'(\|\frac{\partial u}{\partial t}\|_{-1}^{2}+\epsilon \|\frac{\partial u}{\partial t}\|_{{{H}^{1}}(\Omega )}^{2}+\|\frac{\partial \alpha }{\partial t}\|_{{{H}^{2}}(\Omega )}^{2}), \quad c>0. \\ \end{align} | (3.12) |
together with the boundary condition
\dfrac{\partial }{\partial t}\dfrac{\partial u}{\partial t}-\Delta A_{k}\dfrac{\partial u}{\partial t}-\Delta B_{k}\dfrac{\partial u}{\partial t}-\Delta (f'(u)\dfrac{\partial u}{\partial t})=-\Delta(\Delta\dfrac{\partial \alpha}{\partial t}+\Delta\alpha-\dfrac{\partial u}{\partial t}), | (3.13) |
We multiply (3.11) by (-\Delta)^{-1}\dfrac{\partial u}{\partial t} and obtain, owing to (2.9) and the interpolation inequality (3.3),
\mathcal{D^{\beta}}\dfrac{\partial u}{\partial t}=0\quad on\quad \Gamma, \quad \vert\beta\vert\leqslant k. |
hence, owing to (3.10), the differential inequality
\dfrac{d}{dt}\|\dfrac{\partial u}{\partial t}\|_{-1}^{2}+c\|\dfrac{\partial u}{\partial t}\|_{H^{k}(\Omega)}^{2}\leqslant c'(\|\dfrac{\partial u}{\partial t}\|^{2}+\|\Delta\alpha\|^{2}+\|\Delta\dfrac{\partial \alpha}{\partial t}\|^{2}), \quad c>0, | (3.14) |
Summing finally (3.8), \delta_2 times (3.11) and \delta_3 times (3.14), where \delta_2, \delta_3>0 are small enough, we find a differential inequality of the form
\dfrac{d}{dt}\|\dfrac{\partial u}{\partial t}\|_{-1}^{2}+c\|\dfrac{\partial u}{\partial t}\|_{H^{k}(\Omega)}^{2}\leqslant c'(\|\dfrac{\partial u}{\partial t}\|_{-1}^{2}+\|\alpha\|_{H^{2}(\Omega)}^{2}+\|\dfrac{\partial \alpha}{\partial t}\|_{H^{2}(\Omega)}^{2}), \quad c>0. | (3.15) |
where
\dfrac{dE_{2}}{dt}+c(E_2+\|\dfrac{\partial u}{\partial t}\|_{H^{k}(\Omega)}^{2})\leqslant c', \quad c>0, |
Owing to the continuous embedding H^{2k+1}(\Omega)\subset C (\bar{\Omega}), we deduce that
E_2=E_1+\delta_2(\|\Delta\alpha\|^{2}-2((\dfrac{\partial\alpha} {\partial t}, \Delta\alpha))+2((\Delta\dfrac{\partial\alpha} {\partial t}, \Delta\alpha)))+\delta_3\|\dfrac{\partial u}{\partial t}\|_{-1}^{2}. |
and since
\vert\int_{\Omega}F(u_{0})\mathrm{d}x\vert\leqslant Q(\|u_{0}\|_{H^{2k+1}(\Omega)}) |
we see that (-\Delta)^{-\frac{1}{2}}\dfrac{\partial u}{\partial t}(0)\in L^{2}(\Omega) and
(-\Delta)^{-\frac{1}{2}}\dfrac{\partial u}{\partial t}(0)=-(-\Delta)^{\frac{1}{2}} A_{k}u_{0}-(-\Delta)^{\frac{1}{2}} B_{k}u_{0}-(-\Delta)^{\frac{1}{2}} f(u_{0})+(-\Delta)^{\frac{1}{2}}(\alpha_{1}-\Delta\alpha_{1}), | (3.16) |
Furthermore E_2 satisfies
\|\dfrac{\partial u}{\partial t}(0)\|_{-1}\leqslant Q(\|u_{0}\|_{H^{2k+1}(\Omega)}, \|\alpha_{1}\|_{H^{3}(\Omega)}). | (3.17) |
It thus follows from (3.15), (3.16), (3.17) and Growall's lemma that
E_{2}\geqslant c(\|u\|_{H^{k}(\Omega)}^{2}+\|\dfrac{\partial u}{\partial t}\|_{-1}^{2}+\int_{\Omega}F(u)\mathrm{d}x+\|\alpha\|_{H^{2}(\Omega)}^{2}+\|\dfrac{\partial \alpha}{\partial t}\|_{H^{2}(\Omega)}^{2})-c', \quad c > 0. | (3.18) |
and
\begin{align} & \|u(t)\|_{{{H}^{k}}(\Omega )}^{2}+\|\frac{\partial u}{\partial t}(t)\|_{-1}^{2}+\|\alpha (t)\|_{{{H}^{2}}(\Omega )}^{2}+\|\frac{\partial \alpha }{\partial t}(t)\|_{{{H}^{2}}(\Omega )}^{2} \\ & \le {{e}^{-ct}}Q(\|{{u}_{0}}{{\|}_{{{H}^{2k+1}}(\Omega )}}, \|{{\alpha }_{0}}{{\|}_{{{H}^{2}}(\Omega )}}, \|{{\alpha }_{1}}{{\|}_{{{H}^{3}}(\Omega )}})+{c}', \quad c>0, \quad t0, \\ \end{align} | (3.19) |
r>0 given.
Multiplying next (2.7) by \tilde{A}_{k}u, we find, owing to the interpolation inequality (3.3),
\begin{align} & \int_{t}^{t+r}{\|\frac{\partial u}{\partial t}\|_{{{H}^{k}}(\Omega )}^{2}}\text{d}s \\ & \le {{e}^{-ct}}Q(\|{{u}_{0}}{{\|}_{{{H}^{2k+1}}(\Omega )}}, \|{{\alpha }_{0}}{{\|}_{{{H}^{2}}(\Omega )}}, \|{{\alpha }_{1}}{{\|}_{{{H}^{3}}(\Omega )}})+{c}'(r), \quad c>0, \quad t\ge 0, \\ \end{align} |
hence, since f and F are continuous and owing to (3.18),
\dfrac{d}{dt}\|\tilde{A}_{k}^{\frac{1}{2}}u\|^{2}+c\|u\|_{H^{2k}(\Omega)}^{2}\leqslant c'(\|u\|^{2}+\|f(u)\|^{2}+\|\dfrac{\partial \alpha}{\partial t }\|^{2}+\|\Delta \dfrac{\partial \alpha}{\partial t }\|^{2}), \quad c>0, | (3.20) |
Summing (3.15) and (3.22), we have a differential inequality of the form
\begin{align} & \frac{d}{dt}\|\tilde{A}_{k}^{\frac{1}{2}}u{{\|}^{2}}+c\|u\|_{{{H}^{2k}}(\Omega )}^{2} \\ & \le {{e}^{-{c}'t}}Q(\|{{u}_{0}}{{\|}_{{{H}^{2k+1}}(\Omega )}}, \|{{\alpha }_{0}}{{\|}_{{{H}^{2}}(\Omega )}}, \|{{\alpha }_{1}}{{\|}_{{{H}^{3}}(\Omega )}})+{c}", \quad c, {c}'>0, \quad t\ge 0. \\ \end{align} | (3.21) |
where
\begin{align} & \frac{d{{E}_{3}}}{dt}+c({{E}_{3}}+\|u\|_{{{H}^{2k}}(\Omega )}^{2}+\|\frac{\partial u}{\partial t}\|_{{{H}^{k}}(\Omega )}^{2}) \\ & \le {{e}^{-{c}'t}}Q(\|{{u}_{0}}{{\|}_{{{H}^{2k+1}}(\Omega )}}, \|{{\alpha }_{0}}{{\|}_{{{H}^{2}}(\Omega )}}, \|{{\alpha }_{1}}{{\|}_{{{H}^{3}}(\Omega )}})+{c}", \quad c, {c}'>0, \quad t\ge 0, \\ \end{align} |
satisfies
E_{3}=E_{2}+\|\tilde{A}_{k}^{\frac{1}{2}}u\|^{2} | (3.22) |
In particular, it follows from (3.21)-(3.22) that
E_{3}\geqslant c(\|u\|_{H^{k}(\Omega)}^{2}+\|\dfrac{\partial u}{\partial t}\|_{-1}^{2}+\int_{\Omega}F(u)\mathrm{d}x+\|\alpha\|_{H^{2}(\Omega)}^{2}+\|\dfrac{\partial \alpha}{\partial t}\|_{H^{2}(\Omega)}^{2})-c', \quad c > 0. | (3.23) |
r >0 given.
We now multiply (2.7) by u and obtain, employing (2.9) and the interpolation inequality (3.3)
\int_t^{t+r}\|u\|_{H^{2k}(\Omega)}^{2}\mathrm{d}s \leqslant e^{-ct}Q(\|u_{0}\|_{H^{2k+1}(\Omega)}, \|\alpha_0\|_{H^{2}(\Omega)}, \|\alpha_1\|_{H^{3}(\Omega)})+c'(r), \quad c>0, \quad t\geqslant 0, |
whence, proceeding as above,
\dfrac{d}{dt}\|u\|^{2}+c\|u\|_{H^{k+1}(\Omega)}^{2}\leqslant c'(\|u\|_{H^{1}(\Omega)}^{2}+\|\dfrac{\partial \alpha}{\partial t}\|^{2}+\|\Delta \dfrac{\partial \alpha}{\partial t}\|^{2}), \quad c > 0, | (3.24) |
We also multiply (2.7) by \dfrac{\partial u}{\partial t} and find
\dfrac{d}{dt}\|u\|^{2}+c\|u\|_{H^{k+1}(\Omega)}^{2}\leqslant e^{-c't}Q(\|u_{0}\|_{H^{2k+1}(\Omega)}, \|\alpha_0\|_{H^{2}(\Omega)}, \|\alpha_1\|_{H^{3}(\Omega)})+c'', \quad c, c'>0. |
where
\dfrac{d}{dt}(\|\bar{A}_{k}^{\frac{1}{2}}u\|^{2}+\bar{B}_{k}^{\frac{1}{2}}\left[u\right])+c\|\dfrac{\partial u}{\partial t}\|^{2}\leqslant c'\|\Delta f(u)\|^{2}-2((\Delta\dfrac{\partial u} {\partial t}, \dfrac{\partial \alpha}{\partial t}-\Delta \dfrac{\partial \alpha}{\partial t})), |
Since f is of class C^{2}, it follows from the continuous embedding H^{2}(\Omega)\subset C (\bar{\Omega}) that
\bar{B}_{k}^{\frac{1}{2}}\left[u\right]=\sum_{i=1}^{k-1}\sum_{\vert\beta\vert=i}a_{\beta}\|\nabla\mathcal{D^{\beta}}u\|^{2}. |
hence, owing to (3.18),
\|\Delta f(u)\|^{2}\leqslant Q(\|u\|_{H^{2}(\Omega)}), | (3.25) |
Multiply next (2.2) by -\Delta (\dfrac{\partial \alpha}{\partial t}-\Delta \dfrac{\partial \alpha}{\partial t}), we have
\begin{align} & \frac{d}{dt}(\|\bar{A}_{k}^{\frac{1}{2}}u{{\|}^{2}}+\bar{B}_{k}^{\frac{1}{2}}\left[u \right])+c\|\frac{\partial u}{\partial t}{{\|}^{2}} \\ & \le {{e}^{-{c}'t}}Q(\|{{u}_{0}}{{\|}_{{{H}^{2k+1}}(\Omega )}}, \|{{\alpha }_{0}}{{\|}_{{{H}^{2}}(\Omega )}}, \|{{\alpha }_{1}}{{\|}_{{{H}^{3}}(\Omega )}})-2((\Delta \frac{\partial u}{\partial t}, \frac{\partial \alpha }{\partial t}-\Delta \frac{\partial \alpha }{\partial t}))+{c}'', \quad c, {c}'>0. \\ \end{align} | (3.26) |
(note indeed that \|\nabla\dfrac{\partial \alpha}{\partial t}\|^{2}+2\|\Delta\dfrac{\partial \alpha}{\partial t}\|^{2}+\|\nabla\Delta\dfrac{\partial \alpha}{\partial t}\|^{2}=\|\nabla\dfrac{\partial \alpha}{\partial t}-\nabla\Delta\dfrac{\partial \alpha}{\partial t}\|^{2}).
Summing (3.25) and (3.26), we obtain
\begin{align} & \frac{d}{dt}(\|\Delta \alpha {{\|}^{2}}+\|\nabla \Delta \alpha {{\|}^{2}}+\|\nabla \frac{\partial \alpha }{\partial t}-\nabla \Delta \frac{\partial \alpha }{\partial t}{{\|}^{2}})+c(\|\Delta \frac{\partial \alpha }{\partial t}{{\|}^{2}}+\|\nabla \Delta \frac{\partial \alpha }{\partial t}{{\|}^{2}}) \\ & \le 2((\Delta \frac{\partial u}{\partial t}, \frac{\partial \alpha }{\partial t}-\Delta \frac{\partial \alpha }{\partial t})), \quad c>0 \\ \end{align} | (3.27) |
Summing finally (3.21), (3.24) and (3.27), we find a differential inegality of the form
\begin{align} & \frac{d}{dt}(\|\bar{A}_{k}^{\frac{1}{2}}u{{\|}^{2}}+\bar{B}_{k}^{\frac{1}{2}}\left[u \right]+\|\Delta \alpha {{\|}^{2}}+\|\nabla \Delta \alpha {{\|}^{2}}+\|\nabla \frac{\partial \alpha }{\partial t}-\nabla \Delta \frac{\partial \alpha }{\partial t}{{\|}^{2}}) \\ & +c(\|\frac{\partial u}{\partial t}{{\|}^{2}}+\|\Delta \frac{\partial \alpha }{\partial t}{{\|}^{2}}+\|\nabla \Delta \frac{\partial \alpha }{\partial t}{{\|}^{2}}) \\ & \le {{e}^{-{c}'t}}Q(\|{{u}_{0}}{{\|}_{{{H}^{2k+1}}(\Omega )}}, \|{{\alpha }_{0}}{{\|}_{{{H}^{2}}(\Omega )}}, \|{{\alpha }_{1}}{{\|}_{{{H}^{3}}(\Omega )}})+{c}'', \quad c, {c}'>0. \\ \end{align} | (3.28) |
where
\begin{align} & \frac{d{{E}_{4}}}{dt}+c({{E}_{3}}+\|u\|_{{{H}^{k+1}}(\Omega )}^{2}+\|u\|_{{{H}^{2k}}(\Omega )}^{2}+\|\frac{\partial u}{\partial t}{{\|}^{2}}+\|\frac{\partial u}{\partial t}\|_{{{H}^{k}}(\Omega )}^{2}+\|\frac{\partial \alpha }{\partial t}\|_{{{H}^{3}}(\Omega )}^{2}) \\ & \le {{e}^{-{c}'t}}Q(\|{{u}_{0}}{{\|}_{{{H}^{2k+1}}(\Omega )}}, \|{{\alpha }_{0}}{{\|}_{{{H}^{2}}(\Omega )}}, \|{{\alpha }_{1}}{{\|}_{{{H}^{3}}(\Omega )}})+{c}'', \quad c, {c}'>0, \quad t\ge 0 \\ \end{align} |
satisfies, owing to (2.11) and the interpolation inegality (3.3)
E_4=E_3+\|u\|^{2}+\|\bar{A}_{k}^{\frac{1}{2}}u\|^{2}+\bar{B}_{k}^{\frac{1}{2}}\left[u\right]+\|\Delta \alpha\|^{2}+\|\nabla\Delta \alpha\|^{2}+\|\nabla\dfrac{\partial \alpha}{\partial t }-\nabla\Delta \dfrac{\partial \alpha}{\partial t }\|^{2} | (3.29) |
In particular, it follows from (3.28)-(3.29) that
E_{4}\geqslant c(\|u\|_{H^{k+1}(\Omega)}^{2}+\|\dfrac{\partial u}{\partial t}\|_{-1}^{2}+\int_{\Omega}F(u)\mathrm{d}x+\|\alpha\|_{H^{3}(\Omega)}^{2}+\|\dfrac{\partial \alpha}{\partial t}\|_{H^{3}(\Omega)}^{2})-c', \quad c > 0. | (3.30) |
and
\begin{align} & \|u(t){{\|}_{{{H}^{k+1}}(\Omega )}}+\|\alpha (t){{\|}_{{{H}^{3}}(\Omega )}}+\|\frac{\partial \alpha }{\partial t}(t){{\|}_{{{H}^{3}}(\Omega )}} \\ & \le {{e}^{-ct}}Q(\|{{u}_{0}}{{\|}_{{{H}^{2k+1}}(\Omega )}}, \|{{\alpha }_{0}}{{\|}_{{{H}^{3}}(\Omega )}}, \|{{\alpha }_{1}}{{\|}_{{{H}^{3}}(\Omega )}})+{c}', \quad c>0, \quad t\ge 0, \\ \end{align} | (3.31) |
r given.
We finally rewrite (2.7) as an elliptic equation, for t > 0 fixed,
\begin{align} & \int_{t}^{t+r}{(\|}\frac{\partial u}{\partial t}{{\|}^{2}}+\|\frac{\partial \alpha }{\partial t}\|_{{{H}^{3}}(\Omega )}^{2})\text{d}s \\ & \le {{e}^{-ct}}Q(\|{{u}_{0}}{{\|}_{{{H}^{2k+1}}(\Omega )}}, \|{{\alpha }_{0}}{{\|}_{{{H}^{3}}(\Omega )}}, \|{{\alpha }_{1}}{{\|}_{{{H}^{3}}(\Omega )}})+{c}'(r), \quad c>0, \quad t\ge 0, \\ \end{align} | (3.32) |
Multiplying (3.32) by A_{k}u, we obtain, owing to the interpolation inequality (3.3),
A_{k}u=-(-\Delta)^{-1}\dfrac{\partial u}{\partial t}-B_{k}u-f(u)+\dfrac{\partial \alpha}{\partial t}-\Delta \dfrac{\partial \alpha}{\partial t}, \quad\mathcal{D^{\beta}}u=0\quad on\quad \Gamma, \quad \vert\beta\vert\leqslant k-1. |
hence, since f is continuous and owing to (3.18)
\|A_{k}u\|^{2}\leqslant c(\|u\|^{2}+\|f(u)\|^{2}+\|\dfrac{\partial u}{\partial t}\|_{-1}^{2}+\|\dfrac{\partial \alpha}{\partial t}\|^{2}+\|\Delta \dfrac{\partial \alpha}{\partial t}\|^{2}), | (3.33) |
We first have the following theorem.
Theorem 4.1. (i) We assume that (u_{0}, \alpha_{0}, \alpha_{1})\in H_{0}^{k}(\Omega)\times (H^{2}(\Omega)\cap H_{0}^{1}(\Omega))\times (H^{2}(\Omega)\cap H_{0}^{1}(\Omega)), with \int_{\Omega}F (u_{0})\mathrm{d}x < +\infty. Then, (2.1)-(2.4) possesses at last one solution (u, \alpha, \dfrac{\partial \alpha}{\partial t}) such that, \forall T > 0, u (0)=u_{0}, \alpha (0)=\alpha_{0}, \dfrac{\partial \alpha}{\partial t}(0)=\alpha_{1},
\|u(t)\|_{H^{2k}(\Omega)}^{2}\leqslant ce^{-c't}Q(\|u_{0}\|_{H^{2k+1}(\Omega)}, \|\alpha_{0}\|_{H^{3}(\Omega)}, \|\alpha_{1}\|_{H^{3}(\Omega)})+c'', \quad c'>0\quad t\geqslant 0. |
u \in L^{\infty} (\mathbb{R^{+}}; H_{0}^{k}(\Omega))\cap L^{2}(0, T;H^{2k}(\Omega)\cap H_{0}^{k}(\Omega)), |
\dfrac{\partial u}{\partial t}\in L^{\infty} (\mathbb{R^{+}}; H^{-1}(\Omega ))\cap L^{2} (0, T; H_{0}^{k}(\Omega)), |
and
\alpha, \dfrac{\partial \alpha}{\partial t}\in L^{\infty} (\mathbb{R^{+}}; H^{2}(\Omega)\cap H_{0}^{1}(\Omega)) |
\dfrac{d}{dt}((-\Delta)^{-1}u, v))+\sum_{i=1}^{k}\sum_{\vert\beta\vert=i} a_{i}((\mathcal{D^{\beta}}u, \mathcal{D^{\beta}}v))+((f(u), v))=\dfrac{d}{dt}(((u, v)) \\ +((\nabla u, \nabla v))), \forall v\in C_c^{\infty}(\Omega), |
in the sense of distributions.
(ii) If we futher assume that (u_{0}, \alpha_{0}, \alpha_{1})\in (H^{k+1}(\Omega)\cap H_{0}^{k}(\Omega))\times (H^{3}(\Omega)\cap H_{0}^{1}(\Omega))\times (H^{3}(\Omega)\cap H_{0}^{1}(\Omega)), then, \forall T > 0,
\dfrac{d}{dt}(((\dfrac{\partial \alpha}{\partial t}, w))+((\nabla\dfrac{\partial \alpha}{\partial t}, \nabla w))+((\nabla\alpha, \nabla w)))+((\nabla\alpha, \nabla w))=-\dfrac{d}{dt}((u, w)), \forall w\in C_c^{\infty}(\Omega), |
u \in L^{\infty} (\mathbb{R^{+}}; H^{k+1}(\Omega)\cap H_{0}^{k}(\Omega))\cap L^{2} (\mathbb{R^{+}}; H^{k+1}(\Omega)\cap H_{0}^{k}(\Omega)) |
\dfrac{\partial u}{\partial t}\in L^{2} (\mathbb{R^{+}}; L^{2}(\Omega )), |
and
\alpha\in L^{\infty} (\mathbb{R^{+}}; H^{3}(\Omega)\cap H_{0}^{1}(\Omega)) |
The proofs of existence and regularity in (i) and (ii) follow from the a priori estimates derived in the previous section and, e.g., a standard Galerkin scheme.
We then have the following theorem.
Theorem 4.2. The system (1.1)-(1.4) possesses a unique solution with the above regularity.
proof. Let (u^{(1)}, \alpha^{(1)}, \dfrac{\partial\alpha^{(1)}}{\partial t}) and (u^{(2)}, \alpha^{(2)}, \dfrac{\partial\alpha^{(2)}}{\partial t}) be two solutions to (2.1)-(2.3) with initial data (u_{0}^{(1)}, \alpha_{0}^{(1)}, \alpha_{1}^{(1)}) and (u_{0}^{(2)}, \alpha_{0}^{(2)}, \alpha_{1}^{(2)}), respectively. We set
\dfrac{\partial \alpha}{\partial t}\in L^{\infty} (\mathbb{R^{+}}; H^{3}(\Omega)\cap H_{0}^{1}(\Omega))\cap L^{2} (0, T; H^{3}(\Omega)\cap H_{0}^{1}(\Omega)) |
and
(u, \alpha, \dfrac{\partial\alpha}{\partial t})=( u^{(1)}, \alpha^{(1)}, \dfrac{\partial\alpha^{(1)}}{\partial t})-( u^{(2)}, \alpha^{(2)}, \dfrac{\partial\alpha^{(2)}}{\partial t}) |
Then, (u, \alpha) satisfies
( u_{0}, \alpha_{0}, \alpha_{1})=( u_{0}^{(1)}, \alpha_{0}^{(1)}, \alpha_{1}^{(1)})-( u_{0}^{(2)}, \alpha_{0}^{(2)}, \alpha_{1}^{(2)}). | (4.1) |
\dfrac{\partial u}{\partial t}-\Delta A_{k}u-\Delta B_{k}u-\Delta (f(u^{(1)})-f(u^{(2)}))=-\Delta(\dfrac{\partial \alpha}{\partial t}-\Delta\dfrac{\partial \alpha}{\partial t}), | (4.2) |
\dfrac{\partial^{2} \alpha}{\partial t^{2}}-\Delta\dfrac{\partial^{2} \alpha}{\partial t^{2}}-\Delta\dfrac{\partial \alpha}{\partial t}-\Delta \alpha=-\dfrac{\partial u}{\partial t}, | (4.3) |
\mathcal{D^{\beta}} u=\alpha=0\quad\ on \quad\ \Gamma, \quad \vert\beta\vert\leqslant k, | (4.4) |
Multiplying (4.1) by (-\Delta)^{-1}u and integrating over \Omega, we obtain
u|_{t=0}=u_{0}, \alpha|_{t=0}=\alpha_{0}, \dfrac{\partial \alpha}{\partial t}|_{t=0}=\alpha_{1}. |
We note that
\dfrac{d}{dt}\|u\|_{-1}^{2}+c\|u\|_{H^{k}(\Omega)}^{2}\leqslant c'(\|u\|^{2}+\|\dfrac{\partial \alpha}{\partial t}-\Delta\dfrac{\partial \alpha}{\partial t}\|^{2})-2((f(u^{(1)})-f(u^{(2)}, u)). |
with l defined as
f(u^{(1)})-f(u^{(2)})=l(t)u, |
Owing to (2.9), we have
l(t)=\int_0^{1}f'(su^{(1)}(t)+(1-s)u^{(2)}(t))\mathrm{d}s. |
and we obtain owing to the intepolation inequalities (3.3) and (3.10),
\begin{align} & -2((f({{u}^{(1)}})-f({{u}^{(2)}}, u))\le 2{{c}_{0}}\|u{{\|}^{2}} \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \le c\|u{{\|}^{2}} \\ \end{align} | (4.5) |
Next, multiplying (4.2) by (-\Delta)^{-1}(u+\dfrac{\partial \alpha}{\partial t}-\Delta \dfrac{\partial \alpha}{\partial t}), we find
\dfrac{d}{dt}\|u\|_{-1}^{2}+c\|u\|_{H^{k}(\Omega)}^{2}\leqslant c'(\|u\|_{-1}^{2}+\|\dfrac{\partial \alpha}{\partial t}-\Delta\dfrac{\partial \alpha}{\partial t}\|^{2}), \quad c > 0. | (4.6) |
Summing then \delta_4 times (4.5) and (4.6), where \delta_4>0 is small enough, we have, employing once more the interpolation inequality (3.10), a differential inequality of the form
\begin{align} & \frac{d}{dt}(\|\alpha {{\|}^{2}}+\|\nabla \alpha {{\|}^{2}}+\|u+\frac{\partial \alpha }{\partial t}-\Delta \frac{\partial \alpha }{\partial t}\|_{-1}^{2})+c(\|\frac{\partial \alpha }{\partial t}{{\|}^{2}}+\|\frac{\partial \alpha }{\partial t}\|_{{{H}^{1}}(\Omega )}^{2}) \\ & \le {c}'(\|u{{\|}^{2}}+\|\alpha {{\|}^{2}}). \\ \end{align} | (4.7) |
where
\dfrac{dE_{5}}{dt}\leqslant cE_{5}, |
satisfies
E_{5}=\delta_4\|u\|_{-1}^{2}+\|\alpha\|^{2}+\|\nabla \alpha\|^{2}+\|u+\dfrac{\partial \alpha}{\partial t}-\Delta\dfrac{\partial \alpha}{\partial t}\|_{-1}^{2} | (4.8) |
It follows from (4.7)-(4.8) and Gronwall's lemma that
E_{5}\geqslant c(\|u\|_{-1}^{2}+\|\alpha\|_{H^{1}(\Omega)}^{2}+\|\dfrac{\partial \alpha}{\partial t}-\Delta \dfrac{\partial \alpha}{\partial t}\| ^{2}), c>0. | (4.9) |
hence the uniquess, as well as the continuous dependence with respect to the initial data in H^{-1}\times H^{1}\times H^{1}-norm.
All authors declare no conflicts of interest in this paper.
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