Nonlinear Klein-Gordon equation with combined power type nonlinearity and critical initial energy is investigated. The qualitative properties of a new ordinary differential equation are studied and the concavity method of Levine is improved. Necessary and sufficient conditions for finite time blow up and global existence of the solutions are proved. New sufficient conditions on the initial data for finite time blow up, based on the necessary and sufficient ones, are obtained. The asymptotic behavior of the global solutions is also investigated.
Citation: Milena Dimova, Natalia Kolkovska, Nikolai Kutev. Global behavior of the solutions to nonlinear Klein-Gordon equation with critical initial energy[J]. Electronic Research Archive, 2020, 28(2): 671-689. doi: 10.3934/era.2020035
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Nonlinear Klein-Gordon equation with combined power type nonlinearity and critical initial energy is investigated. The qualitative properties of a new ordinary differential equation are studied and the concavity method of Levine is improved. Necessary and sufficient conditions for finite time blow up and global existence of the solutions are proved. New sufficient conditions on the initial data for finite time blow up, based on the necessary and sufficient ones, are obtained. The asymptotic behavior of the global solutions is also investigated.
The aim of this paper is to study the global behavior of the solutions to the Cauchy problem for the nonlinear Klein-Gordon equation
utt−Δu+u=f(u),(t,x)∈R×Rn,u(0,x)=u0(x),ut(0,x)=u1(x),x∈Rn,u0(x)∈H1(Rn),u1(x)∈L2(Rn) | (1) |
with critical initial energy
f(u)=l∑k=1ak|u|pk−1u−s∑j=1bj|u|qj−1u,f(u)=a1|u|p1+l∑k=2ak|u|pk−1u−s∑j=1bj|u|qj−1u, | (2) |
where the constants
a1>0,ak≥0,bj≥0fork=2,…,l,j=1,…,s,1<qs<qs−1<⋯<q1<p1<p2<⋯<pl−1<pl,pl<∞forn=1,2;pl<n+2n−2forn≥3. | (3) |
The combined power type nonlinearity (2) appears in numerous models of quantum mechanics, field theory, nonlinear optics and others. For example, the quadratic-cubic nonlinearity
The global existence or finite time blow up of the solutions to (1) - (3) is fully investigated for nonpositive energy
The case of critical initial energy, i.e.
In the case of supercritical initial energy, i.e.
In our previous paper [4] we prove a necessary and sufficient condition for finite time blow up of the solutions to (1) - (3) for arbitrary positive initial energy
(u(b,⋅),ut(b,⋅))>0and0<E(0)≤p1−12(p1+1)‖u(b,⋅)‖2L2(Rn). | (4) |
Let us emphasize once again that the sign condition
In the present paper we focus on the global behavior of the solutions to (1) - (3) with critical initial energy
The paper is organized in the following way. In Section 2 some preliminary results are given. Section 3 deals with the global behavior of the solutions to a new ordinary differential equation. The results are an improvement of the concavity method of Levine and allow us to formulate necessary and sufficient conditions for finite time blow up. The main results of the paper are formulated and proved in Section 4 and Section 5. In Section 4 the finite time blow up is treated, while Section 5 deals with the global existence of the solutions and their asymptotic behavior.
We will use the following short notations for the functions
‖u‖=‖u(t,⋅)‖L2(Rn),‖u‖1=‖u(t,⋅)‖H1(Rn),(u,v)=(u(t,⋅),v(t,⋅))=∫Rnu(t,x)v(t,x)dx. |
We have the following local existence result to the Cauchy problem (1) - (3), see e.g. [2,6,7].
Theorem 2.1. Problem (1) - (3) admits a unique local weak solution
u(t,x)∈C((0,Tm);H1(Rn))∩C1((0,Tm);L2(Rn))∩C2((0,Tm);H−1(Rn)) |
in the maximal existence time interval
(ⅰ)
iflim supt→Tm,t<Tm‖u‖1<∞,thenTm=∞; |
(ⅱ) for every
E(0)=E(t), | (5) |
where the energy functional
E(t):=E(u(t,⋅),ut(t,⋅))=12(‖ut‖2+‖u‖21)−∫Rn∫u0f(y)dydx. | (6) |
Definition 2.2. The solution
lim supt→Tm,t<Tm‖u‖1=∞. |
In order to prove a necessary and sufficient condition for finite time blow up of the solutions to (1) - (3), we use the following equivalence between the blow up of the
Lemma 2.3. Suppose
lim supt→Tm,t<Tm‖u‖1=∞if and only iflim supt→Tm,t<Tm‖u‖=∞. |
The proof of Lemma 2.3 is based on the Gagliardo - Nirenberg inequality. In one-dimensional case it is given in [4]. The multidimensional case is treated in a similar way and we omit the proof.
Let us recall some important functionals - the Nehari functional
I(t):=I(u(t,⋅))=‖u‖21−∫Rnf(u)udx, | (7) |
J(t):=J(u(t,⋅))=12‖u‖21−∫Rn∫u0f(y)dydx, |
d=infu∈NJ(u(t,⋅)),N={u∈H1(Rn): ‖u‖1≠0, I(u(t,⋅))=0}. | (8) |
In the framework of the potential well method there are two important subsets of
W={u∈H1(Rn):I(u)>0}∪{0},V={u∈H1(Rn):I(u)<0}. |
In the following theorem we formulate the sign preserving properties of
Theorem 2.4. Suppose
(ⅰ) If
(ⅱ) If
Proof. (ⅰ) Suppose
d=E(t0)=12‖ut(t0,⋅)‖2+J(t0)≥d. |
Hence
‖ut(t0,⋅)‖=0,J(t0)=dandI(t0)=0. | (9) |
If
−Δˆu+ˆu−f(ˆu)=0forx∈Rn. |
Consequently, condition (9) means that the function
(ⅱ) Suppose
Remark 1. We rewrite the conservation law (5), (6) by means of (7) in the following way
E(0)=12‖ut‖2+1p1+1I(t)+p1−12(p1+1)‖u‖21+B(t), | (10) |
where from (2) and (3)
B(t)=l∑k=2ak(pk−p1)(pk+1)(p1+1)∫Rn|u|pk+1dx+s∑j=1bj(p1−qj)(qj+1)(p1+1)∫Rn|u|qj+1dx≥0. | (11) |
Remark 2. If
‖u0‖2>2(p1+1)p1−1d−p1+1p1−1‖u1‖2−‖∇u0‖2−2(p1+1)p1−1B(0), | (12) |
while condition
‖u0‖2≤2(p1+1)p1−1d−p1+1p1−1‖u1‖2−‖∇u0‖2−2(p1+1)p1−1B(0). |
For the proofs of our main results in Section 4 and Section 5 we need the following auxiliary statement.
Lemma 2.5. Suppose
I(t)<(p1+1)(J(t)−d)fort∈[0,Tm). | (13) |
The proof of Lemma 2.5 is identical with the proof of Lemma 2.3 in [25] and we omit it.
In the last decades the concavity method, introduced by Levine [15], is one of the powerful methods in the investigation of the finite time blow up of the solutions to nonlinear dispersive equations. The main idea of the concavity method is one to prove finite time blow up of the solutions to the ordinary differential inequality
ψ″(t)ψ(t)−γψ′2(t)≥0,t≥0,γ>1, | (14) |
where
ψ(0)>0,ψ′(0)>0 | (15) |
then the solution
T∗≤ψ(0)(γ−1)ψ′(0). |
In the applications to nonlinear dispersive equations usually
Let us mention, that condition (15) is only sufficient one for finite time blow up of the solution
In order to give a satisfactory answer of this question, instead of inequality (14) we consider the following nonlinear ordinary differential equation
ψ″(t)ψ(t)−γψ′2(t)=Q(t),t∈[0,Tm),0<Tm≤∞,γ>1, | (16) |
Q(t)∈C([0,Tm)),Q(t)≥0,t∈[0,Tm). | (17) |
Here the nonnegative, twice differentiable function
We recall the definition of blow up of a nonnegative function
Definition 3.1. The nonnegative function
lim supt→Tm,t<Tmψ(t)=∞. | (18) |
Theorem 3.2. Suppose
Proof. Step 1. First we will show that
there exists b∈[0,Tm)such thatψ′(b)>0. | (19) |
If not, then
0≤ψ(t)≤ψ(0) |
holds for every
lim supt→Tm,t<Tmψ(t)≤ψ(0), |
which contradicts (18). Thus (19) holds.
Step 2. Now we will prove that
ψ(t)>0fort∈[b,t0)andψ(t0)=0. | (20) |
From (16), (17) and (20) we get
ψ″(t)=(γψ′2(t)+Q(t))ψ−1(t)≥0fort∈[b,t0), |
i.e.
ψ(t)≥ψ(b)>0for everyt∈[b,Tm). | (21) |
Additionally, from (16), (17) and (21) it follows that
ψ′(t)≥ψ′(b)>0for t∈[b,Tm). | (22) |
Step 3. Let us prove that
z(t)=ψ1−γ(t)fort∈[b,Tm). |
Straightforward computations give us
z′(t)=(1−γ)ψ−γ(t)ψ′(t),11−γψ1+γ(t)z″(t)=ψ″(t)ψ(t)−γψ′2(t)≥0 | (23) |
and
z″(t)=−(γ−1)Q(t)zγ+1γ−1(t)fort∈[b,Tm),z(b)>0,z′(b)<0. | (24) |
Suppose that
z″(t)≤0fort≥b. | (25) |
Integrating (25) twice from
z(t)≤z′(b)(t−b)+z(b). |
Consequently, there exists a constant
b<T∗≤b−z(b)z′(b)=b+ψ(b)(γ−1)ψ′(b)<∞, | (26) |
such that
The following necessary and sufficient condition for finite time blow up of the solution to the ordinary differential equation (16) is a key result in the investigation of the behavior of the solutions to nonlinear dispersive equations.
Theorem 3.3. Suppose
Tm≤b+ψ(b)(γ−1)ψ′(b)<∞ | (27) |
holds.
Proof. (Necessity) Suppose
(Sufficiency) Suppose (19) is satisfied. From Step 2 and Step 3 in the proof of Theorem 3.2 it follows that
If we assume by contradiction that
lim supt→Tm,t<Tmψ(t)<∞. | (28) |
From (22), (28) it follows that
limt→Tm,t<Tmψ(t)=ψ0≥0,ψ0<∞. | (29) |
Integrating (24) from
z′(t)=z′(b)−(γ−1)∫tbQ(s)zγ+1γ−1(s)ds, |
or equivalently, from (23)
ψ′(t)=ψγ(t)[ψ′(b)ψγ(b)+(γ−1)2∫tbQ(s)ψ−γ−1(s)ds]. |
Thus from (21), (29) and the monotonicity of
limt→Tm,t<Tmψ′(t)=ψγ0[ψ′(b)ψγ(b)+(γ−1)2∫TmbQ(s)ψ−γ−1(s)ds]=ψ1,0<ψ1<∞. |
The initial value problem
φ″(t)φ(t)−γφ′2(t)=Q(t)fort≥Tm,φ(Tm)=ψ0,φ′(Tm)=ψ1 |
has a classical solution
˜φ(t)={ψ(t)fort∈[0,Tm);φ(t)fort∈[Tm,Tm+δ), |
As a consequence of Theorem 3.2, Theorem 3.3 and Theorem 2.4 we have the following precise results for finite time blow up of the solutions to (1) - (3) in the critical case
Theorem 4.1. Suppose
there existsb∈[0,Tm)such that(u(b,⋅),ut(b,⋅))≥0. | (30) |
Moreover,
Proof. For the function
ψ′(t)=2(u,ut),ψ″(t)=2‖ut‖2−2I(t)=(p1+3)‖ut‖2−2(p1+1)E(0)+(p1−1)‖u‖21+2(p1+1)B(t). | (31) |
Hence
ψ″(t)ψ(t)−p1+34ψ′2(t)=Q(t), | (32) |
where
Q(t)=(p1+3)(‖ut‖2‖u‖2−(u,ut)2)+2{(p1+1)(J(t)−d)−I(t)}‖u‖2+2(p+1)B(t)‖u‖2. | (33) |
From (11), (13) in Lemma 2.5 and the Cauchy-Schwartz inequality we have
Q(t)≥0fort∈[0,Tm). | (34) |
Thus
(Necessity) Suppose
(Sufficiency) Suppose (30) holds, but
ψ′(t)>ψ′(b)≥0for everyt∈(b,Tm). | (35) |
From (35) there exists
Remark 3. From the proof of Theorem 4.1 it is clear that the restriction
Remark 4. Let us compare the condition (4) and the new one (30). The careful analysis of the necessary and sufficient conditions (30) in Theorem 4.1 and (4) shows that if (4) holds then (30) is also satisfied at the same time
In the following theorem we give sufficient conditions for finite time blow up of the solutions to (1) - (3) in terms of the initial data
Theorem 4.2. Suppose
(ⅰ)
(ⅱ)
(u0,u1)<0and‖u0‖2≥2(p1+1)p1−1d−2√p1−1(u0,u1). | (36) |
Moreover,
Proof. (ⅰ) The proof of Theorem 4.2 (ⅰ) follows immediately from the sufficiency part of Theorem 4.1 and Remark 3 for
(ⅱ) Suppose (36) hold. Since
‖u0‖2≥2(p1+1)p1−1d−2√p1−1(u0,u1)>2(p1+1)p1−1d−p1+1p1−1‖u1‖2−‖∇u0‖2−2(p1+1)p1−1B(0) |
from Remark 2 it follows that
In order to prove statement (ⅱ) we suppose by contradiction that
If (30) is satisfied, i.e. there exists
If (30) does not hold, then
ψ′(t)=2(u,ut)<0for everyt≥0. | (37) |
From (31) the function
ψ″(t)=αψ(t)−β+G(t)fort≥0. | (38) |
Here
G(t)=(p1+3)‖ut‖2+(p1−1)‖∇u‖2+2(p1+1)B(t)≥0, |
because
ψ(t)=12(ψ(0)+1√αψ′(0)−βα)e√αt+12(ψ(0)−1√αψ′(0)−βα)e−√αt+βα+1√α∫t0G(s)sinh(√α(t−s))ds | (39) |
and
ψ′(t)=√α2(ψ(0)+1√αψ′(0)−βα)e√αt−√α2(ψ(0)−1√αψ′(0)−βα)e−√αt+∫t0G(s)cosh(√α(t−s))ds. | (40) |
From (39) and (40) we get
ψ(t)+1√αψ′(t)−βα=(ψ(0)+1√αψ′(0)−βα+1√α∫t0G(s)e−√αsds)e√αt | (41) |
By means of (31) the function
h′(t)+εh(t)=I(t)−‖ut‖2−ε(u,ut)=I(t)−(ut+ε2u,ut+ε2u)+ε24‖u‖2=:g(t) |
Since
g(t)≤ε24‖u‖2≤ε24‖u0‖2, |
we have the estimates
h(t)=h(0)e−εt+e−εt∫t0g(s)eεsds≤h(0)e−εt+ε24ε‖u0‖2(1−e−εt). |
After the limit
0≤lim supt→∞h(t)≤ε4‖u0‖2. |
Thus we obtain
limt→∞(u,ut)=0, | (42) |
because
Since
limt→∞ψ(t)−βα=limt→∞(ψ(0)+1√αψ′(0)−βα+1√α∫t0G(s)e−√αsds)e√αt | (43) |
Hence necessarily we have
∫∞0G(s)e−√αsds=−√α(ψ(0)+1√αψ′(0)−βα) |
and from L'Hospital's rule it follows that
limt→∞ψ(t)−βα=limt→∞(ψ(0)+1√αψ′(0)−βα+1√α∫t0G(s)e−√αsds)e√αt=−1αlimt→∞G(t)≤0, | (44) |
i.e.
limt→∞ψ(t)≤βα. |
Multiplying (38) with
ψ′2(t)=α(ψ(t)−βα)2+2∫t0G(s)ψ′(s)ds+K, | (45) |
K=−α(ψ(0)−βα)2+ψ′2(0). |
Since
ψ(0)−βα≥−1√αψ′(0). | (46) |
Thus from (37) it follows that
ψ(0)>βα. | (47) |
Let us consider the case
limt→∞ψ(t)<βα. | (48) |
Since
ψ(t1)=βα. |
Then for
0<ψ′2(t1)=2∫t10G(s)ψ′(s)ds+K<K. | (49) |
Now we consider the case
limt→∞ψ(t)=βα. |
After the limit
0=limt→∞ψ′2(t)=2∫∞0G(s)ψ′(s)ds+K<K. | (50) |
In both cases from (49) and (50) we have
K=−α(ψ(0)−βα)2+ψ′2(0)>0. |
The above inequality is satisfies if
βα+1√αψ′(0)<ψ(0)<βα−1√αψ′(0), |
or equivalently
2(p1+1)p1−1d+2√p1−1(u0,u1)<‖u0‖2<2(p1+1)p1−1d−2√p1−1(u0,u1), |
which contradicts condition (36). Thus
Remark 5. The statement of Theorem 4.2(ⅰ) has been already proved in a different way for the nonlinear wave equation in a bounded domain, see e.g. [9,26] and for nonlinear Klein-Gordon equation, see [16,19]. In the present paper the proof of Theorem 4.2(ⅰ) is a consequence of Theorem 3.3.
Remark 6. Let the initial data satisfy conditions (36). Then from (12) it follows, that
In the following corollary we reformulate the statements in Theorem 4.2. The requirement for the sign of the Nehari functional
Corollary 1. Suppose
(ⅰ)
(u0,u1)≥0and‖u0‖2>2(p1+1)p1−1d−p1+1p1−1‖u1‖2−‖∇u0‖2−2(p1+1)p1−1B(0); |
(ⅱ)
(u0,u1)<0and‖u0‖2≥2(p1+1)p1−1d−2√p1−1(u0,u1). |
Moreover,
Below we compare the result in Theorem 4.2 (Corollary 1) with the result in [16] for the nonlinear term
f(u)=u2+u3. | (51) |
Proposition 1. Suppose
(ⅰ) (Theorem (4.2)(i), [16,Theorem 1.3(3)])
(u0,u1)≥0and‖u0‖2>6d−3‖u1‖2−‖∇u0‖2−12∫Rnu40(x)dx; |
(ⅱ) (Theorem (4.2)(ii))
(u0,u1)<0and‖u0‖2≥6d−2(u0,u1); |
(ⅲ) ([16,Theorem 1.3(3)])
(u0,u1)<0, |
12∫Rnu4e(x)dx+‖∇ue‖2<3‖u1‖2+‖∇u0‖2+12∫Rnu40(x)dx, | (52) |
6d−3‖u1‖2−‖∇u0‖2−12∫Rnu40(x)dx<‖u0‖2≤6d−(12∫Rnu4e(x)dx+‖∇ue‖2), | (53) |
where
Proof. (ⅰ) and (ⅱ) We apply Theorem 4.2 for
B(t)=112∫Rnu4(t,x)dx. |
According to Theorem 4.2 and Corollary 1 the solution
(u0,u1)≥0and‖u0‖2>6d−3‖u1‖2−‖∇u0‖2−12∫Rnu40(x)dx; |
(u0,u1)<0and‖u0‖2≥6d−2(u0,u1), | (54) |
So the statements (ⅰ) and (ⅱ) are proved. Note, that for
(ⅲ) For
(u0,u1)<0,I(0)<0and‖u0‖2≤‖ue‖2, | (55) |
where
d=16‖ue‖21+112∫Rnu4e(x)dx, |
assumptions (55) are equivalent to
(u0,u1)<0,6d−3‖u1‖2−‖∇u0‖2−12∫Rnu40(x)dx<‖u0‖2≤6d−(12∫Rnu4e(x)dx+‖∇ue‖2). |
The above inequality holds only for
12∫Rnu4e(x)dx+‖∇ue‖2<3‖u1‖2+‖∇u0‖2+12∫Rnu40(x)dx. |
When the opposite inequality is satisfied, i.e.
6d−3‖u1‖2−‖∇u0‖2−12∫Rnu40(x)dx≥6d−(12∫Rnu4e(x)dx+‖∇ue‖2), |
then the set of functions satisfying Theorem 1.3(3) in [16] is empty. In this case the finite time blow up of the solutions is possible only under conditions of Theorem (4.2)(ⅱ), i.e. when (54) is satisfied.
However, if (52) holds, then the conditions for finite time of the solution to (1), (51) in Theorem (4.2)(ⅱ) and Theorem 1.3(3) in [16] are completely different. Indeed from the inequality
6d−2(u0,u1)>6d−(12∫Rnu4e(x)dx+‖∇ue‖2), |
it follows that the intervals for
Theorem 5.1. Suppose
(u(t,⋅),ut(t,⋅))<0for everyt≥0. | (56) |
Proof. (Necessity) Suppose
(u(b,⋅),ut(b,⋅))≥0. |
From Theorem 4.1 it follows that
(Sufficiency) Suppose condition (56) is satisfied. Then
‖u(t,⋅)‖2≤‖u0‖2for everyt∈[0,Tm). |
From Lemma 2.3 it follows that
‖u(t,⋅)‖21≤K0<∞for everyt∈[0,Tm) |
and from the local existence result we get
Remark 7. The growth condition
Let us formulate necessary conditions on the initial data for global existence of the solutions to (1) - (3).
Theorem 5.2. Suppose
(ⅰ) If
(ⅱ) If
(u0,u1)<0and‖u0‖2<2(p1+1)p1−1d−2√p1−1(u0,u1). | (57) |
Proof. The statement (ⅰ) in Theorem 5.2 has been already proved in [9,16,24,27] when
(ⅰ) If
d=E(0)=12‖u1‖2+J(u0)≥d, |
i.e.
If
‖u(t,⋅)‖21=2(p1+1)(p1−1)d−(p1+1)(p1−1)‖ut(t,⋅)‖2−2(p1−1)I(t)−2(p1+1)(p1−1)B(t)≤2(p1+1)(p1−1)d<∞. |
Thus from statement (ⅰ) in the local existence result Theorem 2.1 it follows that
(ⅱ) Suppose that
(u0,u1)≥0 |
or
(u0,u1)<0and‖u0‖2≥2(p1+1)p1−1d−2√p1−1(u0,u1), |
then from Theorem 4.2 it follows that
Let us mention that the asymptotic behavior of the global solution to the nonlinear wave equation in bounded domains has been studied in [9] and to the Klein-Gordon equation with quadratic-cubic nonlinearity - in [16].
In the following theorem we study the asymptotic behavior for
Theorem 5.3. Suppose
(ⅰ)
(ⅱ)
(ⅲ)
Proof. Since the solution
‖u(t,⋅‖1≤C0for everyt≥0. | (58) |
Indeed, if (58) fails, then from the local existence result, Theorem 2.1 (ⅰ), we get
|∫Rn∫u0f(y)dydx|≤C1fort≥0 | (59) |
and from the conservation law (6)
‖ut(t,⋅)‖≤C2fort≥0, | (60) |
where the constants
limtm→∞∫Rn∫u(tm,⋅)0f(y)dy Φ(x)dx=∫Rn∫ˆu(x)0f(y)dy Φ(x)dx, |
(u(tm,⋅),Φ)→(ˆu,Φ),(ut(tm,⋅),Θ)→(ˆv,Θ)whentm→∞, | (61) |
i.e.
(ⅰ) We will prove that
0<∫st(‖ut(τ,⋅)‖2−I(τ))dτ=(u(s,⋅),ut(s,⋅))−(u(t,⋅),ut(t,⋅)) | (62) |
0<∫t+1t‖ut(τ,⋅)‖2dτ<∫t+1t(‖ut(τ,⋅)‖2−I(τ))dτ=(u(t+1,⋅),ut(t+1,⋅))−(u(t,⋅),ut(t,⋅)). | (63) |
After the change of the variable
0<∫10‖ut(λ+t,⋅)‖2dλ<(u(t+1,⋅),ut(t+1,⋅))−(u(t,⋅),ut(t,⋅)). | (64) |
Thus for every
limm→∞∫10‖ut(sm+λ,⋅)‖2dλ=0. |
As a consequence of Fatou's lemma we get
lim infm→∞‖ut(sm+λ,⋅)‖2=0fora.e.λ∈[0,1]. | (65) |
By means of the weak convergence of
0≤‖ˆv‖2≤lim infm→∞‖ut(tm,⋅)‖2=0, |
i.e.
‖ˆv‖2=0. |
Thus we proved that every weak limit of
lim inft→∞‖ut(t,⋅)‖=0. |
(ⅱ) From (44) and (38) it follows that
limt→∞ψ(t)−βα+1αlimt→∞G(t)=0,limt→∞ψ″(t)=α(limt→∞ψ(t)+1αlimt→∞G(t)−βα)=0. |
Thus (31) and (ⅰ) in Theorem 5.3 give us
0≥lim supt→∞I(t)≥lim inft→∞I(t)=lim inft→∞‖ut(t,⋅)‖2−12limt→∞ψ″(t)=0. |
Hence
(ⅲ) From (42) and (62) after the limit
∫∞t(‖ut(τ,⋅)‖2−I(τ))dτ=−(u(t,⋅),ut(t,⋅)) | (66) |
and integrating (66) from
∫∞0∫∞t(‖ut(τ,⋅)‖2−I(τ))dτdt=−∫∞0(u(t,⋅),ut(t,⋅))dt=12(‖u0‖2−limt→∞‖u(t,⋅)‖2). | (67) |
Applying Fubini's theorem, (67) becomes
12(‖u0‖2−limt→∞‖u(t,⋅)‖2)=∫∞0∫∞t(‖ut(τ,⋅)‖2−I(τ))dτdt=∫∞0τ(‖ut(τ,⋅)‖2−I(τ))dτ. | (68) |
After the integration of (66) from
∫s0∫∞t(‖ut(τ,⋅)‖2−I(τ))dτdt=12(‖u0‖2−‖u(s,⋅)‖2) |
and Fubini's theorem implies
12(‖u0‖2−‖u(s,⋅)‖2)=∫s0∫τ0(‖ut(τ,⋅)‖2−I(τ))dtdτ+∫∞s∫s0(‖ut(τ,⋅)‖2−I(τ))dtdτ=∫s0τ(‖ut(τ,⋅)‖2−I(τ))dτ+s∫∞s(‖ut(τ,⋅)‖2−I(τ))dτ. | (69) |
After the limit
0=lims→∞s∫∞s(‖ut(τ,⋅)‖2−I(τ))dτ=−lims→∞s(u(s,⋅),ut(s,⋅), |
hence (ⅲ) holds. Theorem 5.3 is proved.
The all authors have been partially supported by the National Scientific Program "Information and Communication Technologies for a Single Digital Market in Science, Education and Security (ICTinSES)", contract No D01205 / 23.11.2018, financed by the Ministry of Education and Science in Bulgaria. The first author has been also supported by the Bulgarian National Science Fund under grant DFNI 12/5. The second author has been also supported by the Bulgarian National Science Fund under grant K
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