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Kyungpook Mathematical Journal 2020; 60(1): 145-161

Published online March 31, 2020

Copyright © Kyungpook Mathematical Journal.

Existence and Behavior Results for a Nonlocal Nonlinear Parabolic Equation with Variable Exponent

Uğur Sert∗, Eylem ÖztÜrk

Department of Mathematics, Hacettepe University, 06800, Beytepe, Ankara, Turkey
e-mail : usert@hacettepe.edu.tr
Department of Mathematics, Hacettepe University, 06800, Beytepe, Ankara, Turkey
e-mail : eozturk@hacettepe.edu.tr

Received: October 12, 2018; Revised: October 4, 2019; Accepted: November 18, 2019

In this article, we study the solvability of the Cauchy-Dirichlet problem for a class of nonlinear parabolic equations with nonstandard growth and nonlocal terms. We prove the existence of weak solutions of the considered problem under more general conditions. In addition, we investigate the behavior of the solution when the problem is homogeneous.

Keywords: degenerate parabolic equations, nonstandard nonlinearity, nonlocal source, solvability theorem, embedding theorems.

This paper deals with the existence and behavior of the solution of a nonlinear parabolic Dirichlet-type boundary value problem whose model example is the following:

{ut-i=1nDi(up0-2Diu)+a(x,t,u)+g(x,t)uLp(Ω)s(t)=h(x,t),u(x,0)=0=u0(x),         uΓT=0

where (x, t) ∈ QT := Ω × (0, T), T > 0, ΓT := Ω × [0, T], Ω ⊂ ℝn (n ≥ 3) is a bounded domain with sufficiently smooth boundary (at least Lipschitz), Di∂/∂xi and p0 ≥ 2, p, s ≥ 1 and a : Ω×(0, T)×ℝ → ℝ, a(x, t, τ) is a function with variable nonlinearity in τ, (for example, a (x, t, τ) = a0 (x, t) |τ |α(x,t)−1 +a1 (x, t)) and g is a real valued measurable function which is not zero on the cylinder QT.

Recently, nonlinear parabolic equations with nonlocal terms have been well studied ([2, 4, 8, 9, 10, 11, 12, 13, 16]). Here, “nonlocal term” denotes a functional depending on the unknown function. There are numerous nonlocal mathematical models studied by many authors to express processes in physics and engineering. For example, Galaktionov and Levine [18] presented a general approach to critical Fujita exponents for nonlinear parabolic problems with nonlocal nonlinearities. Pao [24] considered a nonlocal model obtained from combustion theory. The degenerate parabolic equations with a nonlocal term which appear in a population dynamics model that communicates through chemical means, were studied in [4, 12, 17].

The equation in (1.1) is nonlinear with respect to the solution, and for the case p0 = 2, this equation is a nonlocal reaction-diffusion equation which describes an ignition model for a compressible reactive gas (see, [4, 7]). In this case the existence, uniqueness and blow-up of nonnegative solutions to problem (1.1) have been studied in [23, 27, 34, 35]. Models similar to (1.1) also arise in biology to describe the density of some biological species. In such models the nonlocal term and the absorption term cooperate and communicate during the diffusion.

Boundary-value problems of type (1.1) are a case of the Newtonian filtration equation which can be given in the following general form:

ut=Δϕ(u)+f.

Equation (1.1) is a parabolic equation with implicit degeneracy which is similar to the equation of Newtonian polytropic filtration [15, 19, 21, 22] i.e.

ut=Δ(um-1u)+f,

where m > 1. This equation is parabolic for u different from 0 and degenerates when u = 0. Under the condition m > 1, the above equation describes the non-stationary flow of a compressible Newtonian fluid in a porous medium under polytropic conditions.

Over the past decade, there has been an increasing interest in the study of degenerate parabolic equations that involve variable exponents [3, 5, 6]. In this paper, we investigate the parabolic equation with such an additional term f, together with variable nonlinearity and nonlocal terms. If we rearrange the main part of the equation, we arrive at

ut=Δ(up0-2u)+F(x,t,u,uLp(Ω),h).

To the best of our knowledge, there have not been any studies on the existence of solutions for the parabolic equations of type (1.1) providing a function F whose argument depends on nonlinear nonlocal term uLp(Ω)s(t) and a separate |u| with variable nonlinearity. We stress that the nonlinearity of nonlocal term g(x,t)uLp(Ω)s(t) is independent from the local nonlinearity. This causes some difficulties in studying the uniqueness and behavior of the solution of problem (1.1).

We apply the general solvability theorem of [31], i.e. Theorem 2.6, to prove the existence of weak solution of (1.1). We study problem (1.1) on the domain of the operator generated by the addressed problem and verify the existence of a sufficiently smooth solution of the problem under more general (weak) conditions. Investigating a boundary-value problem on its own space yields better results. Therefore in this work, we analyse the considered problem on its own space. Apart from linear boundary value problems, the sets generated by nonlinear problems are subsets of linear spaces which do not have linear structure (see [28, 29, 30, 31, 32, 33] and references therein).

This paper is organized as follows. In the next section, we recall some useful results on the generalized Orlicz-Lebesgue spaces and results on nonlinear spaces (pn-spaces). In Section 3, we present the assumptions, define the weak solution, and then prove the existence of weak solution to problem (1.1). In Section 4, we examine the behavior of the solution of (1.1) when the problem is homogeneous.

In this section, we begin with some available facts from the theory of the generalized Lebesgue spaces which are also called Orlicz-Lebesgue spaces. We present these facts without proof; proofs can be found in [1, 14, 20, 25].

Let Ω be a Lebesgue measurable subset of ℝn such that |Ω| > 0. (Throughout the paper, we denote by |Ω| the Lebesgue measure of Ω). Let p (x, t) ≥ 1 be a measurable bounded function defined on the cylinder QT = Ω× (0, T) i.e.

1p-essQTinfp(x,t)essQTsupp(x,t)p+<.

On the set of all functions on QT define the functional σp and ||.||p by

σp(u)QTup(x,t)dxdt

and

uLp(x,t)(QT)inf{λ>0σp(uλ)1}.

The Generalized Lebesgue space is defined as follows:

Lp(x,t)(QT):={u:uis a measurable real-valued function in QT,σp(u)<}.

The space Lp(x,t) (QT) becomes a Banach space under the norm ||.||Lp(x,t) (QT) which is so-called Luxemburg norm.

We present the following results for these spaces (see [20, 25, 26]):

Lemma 2.1

If 0 < |Ω| < ∞, and p1and p2fulfill (2.1), then

Lp1(x,t)(QT)Lp2(x,t)(QT)p2(x,t)p1(x,t)for a.e (x,t)QT.

Lemma 2.2

The dual space of Lp(x,t) (QT) is Lp* (x,t) (QT) if and only if pL (QT). The space Lp(x,t) (QT) is reflexive if and only if

1<p-p+<

herep*(x,t)p(x,t)p(x,t)-1.

For uLp(x,t) (QT) and vLq(x,t) (QT) where p, q satisfy (2.1) and 1p(x,t)+1q(x,t)=1, the following inequalities hold:

QTuvdxdt2uLp(x,t)(QT)vLq(x,t)(QT)

and for all uLp(x,t) (Ω), we have

min{uLp(x,t)(QT)p-,uLp(x,t)(QT)p+}σp(u)max{uLp(x,t)(QT)p-,uLp(x,t)(QT)p+}.

We introduce certain nonlinear function spaces (pn-spaces) which are complete metric spaces and directly connected to the problem under consideration. We also give some embedding results for these spaces [33, 32, 30, 31] (see also references cited therein).

Definition 2.3

Let α ≥ 0, β ≥ 1, ϱ = (ϱ1,...,ϱn) be multi-index, m ∈ ℤ+ and Ω ⊂ ℝn (n ≥ 1) be bounded domain with sufficiently smooth boundary.

Sm,α,β(Ω){uL1(Ω)[u]Sm,α,β(Ω)α+β0ϱm(ΩuαDϱuβdx)<}

in particular,

S°1,α,β(Ω){uL1(Ω)[u]S°1,α,β(Ω)α+βi=1n(ΩuαDiuβdx)<}{uΩ0}

and for p ≥ 1,

Lp(0,T;S°1,α,β(Ω)){uL1(QT)[u]Lp(0,T;S°1,α,β(Ω))p0T[u]S°1,α,β(Ω)pdt<}.

These spaces are called pn-spaces.*

Theorem 2.4

Let α ≥ 0, β ≥ 1 then ϕ : ℝ → ℝ, ϕ(t)tαβt is a homeomorphism between S1,α,β (Ω) and W1 (Ω).

Theorem 2.5

The following embeddings hold:

  • Let α, α1 ≥ 0 and β1 ≥ 1, ββ1, α1β1αβ, α1 + β1α + β then we have

    S°1,α,β(Ω)S°1,α1,β1(Ω).

  • Let α ≥ 0, β ≥ 1, n > β andn(α+β)n-βrthen there is a continuous embedding

    S°1,α,β(Ω)Lr(Ω).

    Furthermore forn(α+β)n-β>rthe embedding is compact.

  • If α ≥ 0, β ≥ 1 and pα + β then

    W01,p(Ω)S°1,α,β(Ω)

    holds.

In the following, we present the general solvability theorem of [31], whose proof relies on Galerkin approximation (see also for similar theorems [33, 30]). We will employ this theorem to demonstrate the existence of a weak solution of problem (1.1).

Theorem 2.6

Let X and Y be Banach spaces with dual spaces X*and Y*respectively, Y be a reflexive Banach space, M0⊆ X be a weakly complete “reflexive” pn-space, X0⊆ M0Y be a separable vector topological space. Let the following conditions be fulfilled:

  • f : S0Lq (0, T; Y) is a weakly compact (weakly continuous) mapping, where

    S0:=Lp(0,T;M0)W1,q(0,T;Y){x(t):x(0)=0}

    1 < max {q, q′} ≤ p < ∞, q=qq-1;

  • there is a linear continuous operator A : Ws,m (0, T;X0) → Ws,m (0, T; Y*), s ≥ 0, m ≥ 1 such that A commutes withtand the conjugate operator A*has ker(A*) = 0;

  • operators f and A generate, in generalized sense, a coercive pair on space Lp (0, T;X0), i.e. there exist a number r > 0 and a functionΨ:+1+1such that Ψ(τ) ↗ ∞ as τ ↗ ∞ and for any xLp (0, T;X0) such that [x]Lp (M0)r following inequality holds:

    0Tf(t,x(t)),Ax(t)dtΨ([x]Lp(M0));

  • there exists some constants C0 > 0, C1, C2 ≥ 0 and ν > 1 such that the inequalities

    0Tξ(t),Aξ(t)dtC0ξLq(0,T;Y)ν-C2,0txτ,Ax(τ)dτC1xYν(t)-C2,         a.e.   t[0,T]

    hold for any xW1,p (0, T;X0) and ξLp (0, T;X0).

Assume that that conditions (i)–(iv) are fulfilled. Then the Cauchy problem

dxdτ+f(t,x(t))=y(t),         yLq(0,T;Y);         x(0)=0

is solvable in S0in the following sense

0Tdxdτ+f(t,x(t)),y*(t)dt=0Ty(t),y*(t),         y*Lq   (0,T;Y*),

for any yLq (0, T; Y) satisfying the inequality

sup{1[x]Lp(0,T;M0)0Ty(t),Ax(t)dt:xLp(0,T;X0)}<.

Let Ω ⊂ ℝn (n ≥ 3) be a bounded domain with sufficiently smooth boundary Ω. We study the problem

{ut-i=1nDi(up0-2Diu)+a(x,t,u)+g(x,t)uLp(Ω)s(t)=h(x,t),         (x,t)QTu(x,0)=0=u0(x),         uΓT=0

under the following conditions:

p0 ≥ 2, p,s ≥ 1, g : QT → ℝ is a measurable function satisfying g(x, t) ≠ 0 for a.e. (x, t) ∈ QT and a : Ω × (0, T) × ℝ → ℝ, a(x, t, τ) is a Carathédory function with variable nonlinearity in τ (see inequality (3.1)).

Let the function a (x, t, τ) in problem (1.1) fulfill the following conditions:

(U1)There exists a measurable function α : Ω × (0, T) → ℝ, 1 < αα (x, t) ≤ α+ < ∞ such that a (x, t, τ) satisfies the inequalities

a(x,t,τ)a0(x,t)τα(x,t)-1+a1(x,t)

and

a(x,t,τ)τa2(x,t)τα(x,t)-a3(x,t),

a.e. (x, t, τ) ∈ QT × ℝ.

Here ai, i = 0, 1, 2, 3 are nonnegative, measurable functions defined on QT and a2 (x, t) ≥ A0 > 0 a.e. (x, t) ∈ QT.

We investigate problem (1.1) for the functions hLq0 (0, T;W−1,q0 (Ω)) + Lα* (x,t) (QT) where α* is conjugate of α i.e. α*(x,t):=α(x,t)α(x,t)-1

and the dual space W-1,q0(Ω):=(W01,p0(Ω))*,q0:=p0p0-1..

Let us denote S0 by

S0:=Lp0(0,T;S°1,(p0-2)q0,q0(Ω))Lα(x,t)(QT)W1,q0(0,T;W-1,q0(Ω)){u:u(x,0)=0}.

We understand the solution of the problem under consideration in the following sense:

Definition 3.1

A function uS0, is called the generalized solution (weak solution) of problem (1.1) if it satisfies the equality

0TΩutwdxdt+i=1n0TΩ(up0-2Diu)Diwdxdt+0TΩa(x,t,u)wdxdt+0TΩg(x,t)uLp(Ω)swdxdt=0TΩhwdxdt

for all wLp0(0,T;W01,p0(Ω))Lα(x,t)(QT)W1,q0(0,T;W-1,q0(Ω)).

We are ready to proceed to the main theorem of this section but first define the followings. For sufficiently small η ∈ (0, 1)

Q1,T:={(x,t)QTα(x,t)[1,p0-η)},Q2,T:={(x,t)QTα(x,t)[p0-η,α+]}

and

β(x,t):={p0α*(x,t)p0-α(x,t)if   (x,t)Q1,T,if   (x,t)Q2,T.

Also, p0˜:=np0n-q0 which is critical exponent in Theorem 2.5 and its conjugate is p0˜*=p0˜p0˜-1.

Theorem 3.2. (Existence Theorem)

Let(U1) be satisfied; 1 ≤ s < p0 − 1 and pp0. If a0Lβ(x,t) (QT), a1Lα* (x,t) (QT), a2L (QT), a3L1 (QT) andgLp0p0-(s+1)(0,T;Lp0˜*(Ω))then for all hLq0 (0, T;W−1,q0 (Ω))+Lα* (x,t) (QT) problem(1.1)has a generalized solution in the space S0and ∂u/∂t belongs to Lq0 (0, T;W−1,q0 (Ω)).

The proof of Theorem 3.2 is based on the general existence theorem (Theorem 2.6). We introduce the following spaces and mappings in order to apply Theorem 2.6 to prove Theorem 3.2.

S0:=Lp0(0,T;S°1,(p0-2)q0,q0(Ω))Lα(x,t)(QT)W1,q0(0,T;W-1,q0(Ω)){u:u(x,0)=0},f:S0Lq0(0,T;W-1,q0(Ω))+Lα*(x,t)(QT),f(u):=-i=1nDi(up0-2Diu)+a(x,t,u)+g(x,t)uLp(Ω)s(t),A:Lp0(0,T;W01,p0(Ω))Lα(x,t)(QT)S0Lp0(0,T;W01,p0(Ω))Lα(x,t)(QT),A:=Id.

We prove some lemmas to show that all conditions of Theorem 2.6 are fulfilled under the conditions of Theorem 3.2.

Lemma 3.3

Under the conditions of Theorem 3.2, f and A generate a “coercive pair” onLp0(0,T;W01,p0(Ω))Lα(x,t)(QT).

Proof

Since AId, being “coercive pair” equals to order coercivity of f on the space Lp0(0,T;W01,p0(Ω))Lα(x,t)(QT).

For uLp0(0,T;W01,p0(Ω))Lα(x,t)(QT), we have the following equation:

f(u),uQT=i=1n(0TΩup0-2Diu2dxdt)+QTa(x,t,u)udxdt+0TΩg(x,t)uLp(Ω)sudxdt.

By using (3.2), we obtain

f(u),uQTi=1n(0TΩup0-2Diu2dxdt)+QTa2(x,t)uα(x,t)dxdt-QTa3(x,t)dxdt-0TΩg(x,t)uLp(Ω)sudxdt.

If we employ (U1) to estimate the second integral in (3.3) and by applying Hölder inequality together with the embedding 1,(p0−2)q0,q0 (Ω) ⊂ Lp (Ω) (see Theorem 2.5) to estimate the fourth integral then we get,

f(u),uQT[u]Lp0(0,T;S°1,(p0-2),2(Ω))p0+A0QTuα(x,t)dxdt-C0T[u]S°1,(p0-2)q0,q0(Ω)suLp0˜(Ω)gLp0˜*(Ω)dt-a3L1(QT).

By taking account the embeddings (see Theorem 2.5)

S°1,(p0-2),2(Ω)S°1,(p0-2)q0,q0(Ω)

and

S°1,(p0-2)q0,q0(Ω)Lp0˜(Ω)

into (3.4) to estimate the pseudo-norm and third integral respectively, we attain

f(u),uQTC0[u]Lp0(0,T;S°1,(p0-2)q0,q0(Ω))p0+A0QTuα(x,t)dxdt-C10T[u]S°1,(p0-2)q0,q0(Ω)s+1gLp0˜*(Ω)dt-a3L1(QT).

By utilizing Young’s inequality to the third integral in (3.5), we have

f(u),uQTC2([u]Lp0(0,T;S°1,(p0-2)q0,q0(Ω))p0+uLα(x,t)(QT)α-)-K.

Here, K=K(a3L1(QT),gLp0p0-(s+1)(0,T;Lp0˜*(Ω))), C2 = C2 (p0, s, A0, |Ω|) are positive constants. So the proof is completed.

Lemma 3.4

Under the conditions of Theorem 3.2, f is bounded from S0into Lq0 (0, T;W−1,q0 (Ω))+ Lα* (x,t) (QT).

Proof

First, we define the mappings

f1(u):=i=1n-Di(up0-2Diu)+g(x,t)uLp(Ω)s(t),f2(u):=a(x,t,u).

We need to show that these mappings are both bounded from Lp0 (0, T; 1,(p0−2)q0,q0 (Ω)) ∩ Lα(x,t) (QT) into Lq0 (0, T; W−1,q0 (Ω)) + Lα*(x,t) (QT).

Let us show that f1 is bounded: For uLp0 (0, T; 1,(p0−2)q0,q0 (Ω)) and vLp0(0,T;W01,p0(Ω)),

f1(u),vQTi=1n(0TΩup0-2DiuDivdxdt)+0TΩg(x,t)uLp(Ω)svdxdt.

Using the embedding 1,(p0−2)q0,q0 (Ω) ⊂ Lp (Ω) and Hölder’s inequality above we find,

[i=1n(0TΩu(p0-2)q0Diuq0dxdt)]1q0[i=1n(0TΩDivp0dxdt)]1p0+C˜0T[u]S°1,(p0-2)q0,q0(Ω)sgLnp0n(p0-1)+p0(Ω)vW01,p0(Ω)dt.

Estimating the second integral above by Hölder’s inequality (p0s>1), we obtain

f1(u),vQTΨ([u]Lp0(0,T;S°1,(p0-2)q0,q0(Ω)))vLp0(0,T;W01,p0(Ω))

where

Ψ([u]Lp0(0,T;S°1,(p0-2)q0,q0(Ω)))=[u]Lp0(0,T;S°1,(p0-2)q0,q0(Ω))p0-1+C˜1[u]Lp0(0,T;S°1,(p0-2)q0,q0(Ω))sgLp0p0-(s+1)(0,T;Lp0˜*(Ω)).

By the last inequality, boundedness of f1 is achieved.

Similarly, from (3.1) and Theorem 2.5, for all uS0, we have the following estimate

σα*(f2(u))=σα*(a(x,t,u))=0TΩa(x,t,u)α*(x,t)dxdtC3(σα(u)+[u]Lp0(0,T;S°1,(p0-2)q0,q0(Ω))p0)+C4,

here C3 = C3 (α+, α, ||a0||Lβ(x,t) (QT)), C4 = C4 (σβ (a0), σα* (a1), |Ω|) > 0 are constants. That yields f2 : Lp0 (0, T; 1,(p0−2)q0,q0 (Ω)) ∩ Lα(x,t) (QT) → Lα* (x,t) (QT) is bounded.

Lemma 3.5

Under the conditions of Theorem 3.2, f is weakly compact from S0into Lq0 (0, T;W−1,q0 (Ω))+ Lα* (x,t) (QT).

Proof

First we verify the weak compactness of f0, where f0(u):=-i=1nDi(up0-2Diu). Let {um(x,t)}m=1S0 be bounded and umS0u0. It is sufficient to show a subsequence of {umj}m=1{um}m=1 which satisfies f0(umj)Lq0(0,T;W-1,q0(Ω))f0(u0).

Since for a.e. t ∈ (0, T), um (·, t) ∈ 1,(p0−2)q0,q0 (Ω), and by existence of an one-to-one correspondence between the classes (Theorem 2.4)

S°1,(p0-2)q0,q0(Ω)ϕ-1ϕW01,q0(Ω)

with the homeomorphism

ϕ(τ)τp0-2τ,ϕ-1(τ)τ-p0-2p0-1τ,

for all m ≥ 1 we have

ump0-2umLq0(0,T;W01,q0(Ω))

is bounded. Due to the fact Lq0(0,T;W01,q0(Ω)) is a reflexive space, there exists a subsequence {umj}m=1{um}m=1 such that

umjp0-2umjLq0(0,T;W01,q0(Ω))ξ.

Now, we show that ξ = |u0|p0−2u0. According to compact embedding [33],

Lp0(0,T;S°1,(p0-2)q0,q0(Ω))W1,q0(0,T;W-1,q0(Ω))Lp0(QT){umjk}m=1{umj}m=1,umjkLp0(QT)u0

which implies

umjka.eQTu0

by the continuity of ϕ(τ), we get

umjkp0-2umjka.eQTu0p0-2u0

that yields ξ = |u0|p0−2u0.

From this, we deduce that for each vLp0(0,T;W01,p0(Ω))

f0(umjk),vQT=i=1n-Di(umjkp0-2Diumjk),vQTmji=1n-Di(u0p0-2Diu0),vQT=f0(u0),vQT

whence, the result is obtained.

We shall show the weak compactness of f2. Since

a:Lp0(0,T;S°1,(p0-2)q0,q0(Ω))Lα(x,t)(QT)Lα*(x,t)(QT)

is bounded by Lemma 3.4, then for m ≥ 1, f2(um)={a(x,t,um)}m=1Lα*(x,t)(QT). Also Lα* (x,t) (QT) (1 < (α*) < ∞) is a reflexive space thus {um}m=1 has a subsequence {umj}m=1 such that

a(x,t,umj)Lα*(x,t)(QT)ψ.

We deduce from the compact embedding (3.6) that

{umjk}m=1{umj}m=1,umjkLp0(QT)u0

thus

umjka.eQTu0.

Accordingly, the continuity of a (x, t, .) for almost (x, t) ∈ QT implies that

a(x,t,umjk)a.eQTa(x,t,u0),

so, we arrive at ψ = a (x, t, u0) i.e. f2(umjk)Lq0(0,T;W-1,q0(Ω))+Lα*(x,t)(QT)f2(u0).

Let a1 (x, t, u) := g (x, t) ||u||Lp (Ω) (t). Using the compact imbedding (3.6) and pp0, we attain

g(x,t)umjLp(Ω)sLq0(0,T;W-1,q0(Ω))g(x,t)u0Lp(Ω)s(t).

Therefore, a1 is weakly compact from S0 into Lq0 (0, T;W−1,q0 (Ω))+Lα*(x,t) (QT). As a conclusion, f is weakly compact from S0 into Lq0 (0, T;W−1,q0 (Ω)) + Lα*(x,t) (QT).

Now, we give the proof of main theorem of this section.

Proof of Theorem 3.2

Since A = Id, obviously it is a linear bounded map and satisfies the conditions (ii) of Theorem 2.6. Furthermore for any uW01,p0(QT) the following inequalities are valid:

0Tu,uΩdt=0TuL2(Ω)2dtMuLq0(0,T;W-1,q0(Ω))2

and

0tuτ,uΩdτ=12uL2(Ω)2(t)M12uW-1,q0(Ω)2(t),

a.e. t ∈ [0, T] (constant M >0 comes from embedding inequality). Thus condition (iv) of Theorem 2.6 is satisfied as well. Consequently from Lemma 3.3-Lemma 3.5, it follows that the mappings f and A fulfill all the conditions of Theorem 2.6. Employing this theorem to problem (1.1), we find that (1.1) is solvable in S0 for any hLq0 (0, T;W−1,q0 (Ω))+ Lα*(x,t) (QT) satisfying the following inequality

sup{1[u]Lp0(0,T;S°1,(p0-2)q0,q0(Ω))+uLα(x,t)(QT)0Th,uΩdt:uQ0}<

where Q0:=Lp0(0,T;W01,p0(Ω))Lα(x,t)(QT). Considering the norm definition of h in Lq0 (0, T;W−1,q0 (Ω))+Lα*(x,t) (QT), we conclude that (1.1) is solvable in S0 for any hLq0 (0, T;W−1,q0 (Ω))+ Lα*(x,t) (QT). In order to complete the proof, it remains to remark that (1.1) can be written in the form

ut=h(x,t)-F(x,t,u,Diu),

and under the conditions of Theorem 3.2, right hand belongs to Lq0 (0, T;W−1,q0 (Ω)) which implies ∂u/∂tLq0 (0, T;W−1,q0 (Ω)).

Remark 3.6

We note that if the function α (x, t) in (3.1) satisfies the inequality α+ < p0 then the existence of a solution of the problem (1.1) can be shown under more general (weaker) conditions. This is verified in the following theorem.

Theorem 3.7

Assume that (3.1) and inequalities 1 ≤ s < p0 − 1, pp0are satisfied. If 1 < αα (x, t) ≤ α+ < p0, (x, t) ∈ QT andgLp0p0-(s+1)(0,T;Lp0˜*(Ω)), a0Lβ1(x,t) (QT), a1Lα*(x,t) (QT) whereβ1(x,t):=p0α*(x,t)p0-α(x,t)thenhLq0 (0, T;W−1,q0 (Ω)) problem(1.1)has a generalized solution in the space Lp0 (0, T; 1,(p0−2)q0,q0 (Ω)) ∩ W1,q0 (0, T;W−1,q0 (Ω)).

Proof

We deduce from inequality (3.1) that

f(u),uQTi=1n(0TΩup0-2Diu2dxdt)-QTa0(x,t)uα(x,t)dxdt-QTa1(x,t)dxdt-0TΩg(x,t)uLp(Ω)sudxdt.

For arbitrary ε > 0 estimating the second integral above by Young’s inequality and using Lp0 (0, T; 1,(p0−2)q0,q0 (Ω)) ⊂ Lp0 (QT), we attain the following inequality which gives the coercivity of f,

f(u),uQTC5[u]Lp0(0,T;S°1,(p0-2)q0,q0(Ω))p0-K˜.

here C5 = C5 (p0, |Ω|, s) and

K˜=K˜(ɛ,a0Lβ1(x,t)(QT),a1Lα*(x,t)(QT),gLp0p0-(s+1)(0,T;Lp0˜*(Ω))).

By the embedding

Lp0(0,T;S°1,(p0-2)q0,q0(Ω))Lp0(QT)Lα(x,t)(QT),

weak compactness and boundedness of f : Lp0 (0, T; 1,(p0−2)q0,q0 (Ω)) ∩ W1,q0 (0, T;W−1,q0 (Ω)) → Lq0 (0, T;W−1,q0 (Ω)) follows from Lemma 3.4 and Lemma 3.5. Thus by the virtue of the proof of Theorem 3.2, we get the desired result.

In this section, we analyze problem (1.1) in homogeneous case. We establish sufficient conditions which ensure that problem (1.1) has only trivial solution under these conditions.

Theorem 4.1

Let conditions of Theorem 3.2 be fulfilled with the following assumptions:

  • Let h(x, t) = 0 and p = 2, p0 > 2.

  • Condition (3.2) is satisfied with a3(x, t) = 0.

  • The functional ||g||L2 (Ω)(t) is bounded for almost every t ∈ ℝ+,

then problem(1.1)has only trivial solution.

Proof

Conditions of Theorem 4.1 provide that (1.1) has a solution in S0. It follows from Definition 3.1 that every weak solution satisfies the following relation,

12ddtuL2(Ω)2+i=1nΩ(up0-2(Diu)2dx)+Ωa(x,t,u)udx+Ωg(x,t)uL2(Ω)sudx=0

then we get

12ddtuL2(Ω)2+4p02i=1nΩ(Di(up02))2dx+Ωa(x,t,u)udx+Ωg(x,t)uL2(Ω)sudx=0

by using imbedding inequality and condition (ii), we deduce that

12ddtuL2(Ω)2+4p02cΩ(up0dx)+Ωg(x,t)uL2(Ω)sudx0

employing Hölder inequality and condition (iii) to the last term we find that

12ddtuL2(Ω)2+4p02cΩup0dx-KuL2(Ω)s+10

where K >0 is a constant. From embedding inequality, we obtain

12ddtuL2(Ω)2+4p02cΩp0-22uL2(Ω)p0-KuL2(Ω)s+10

whence denoting by y=uL2(Ω)2 and μ=p02, we have

12dydt+4p02cΩp0-22yμ-Kys+120

by utilizing Young inequality to the last term in the above equation, we attain

12dydt+(4p02cΩp0-22-Kɛ)yμ-Kc(ɛ)y0

where ɛ<4Kp02cΩp0-22 from here, we conclude

12dydtKc(ɛ)y.

Integrating the last inequality and considering y(0) = 0, we arrive at the desired result.

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