IJDE International Journal of Differential Equations 1687-9651 1687-9643 Hindawi 10.1155/2018/9754567 9754567 Research Article Well-Posedness and Numerical Study for Solutions of a Parabolic Equation with Variable-Exponent Nonlinearities Al-Smail Jamal H. 1 http://orcid.org/0000-0003-1061-0075 Messaoudi Salim A. 1 Talahmeh Ala A. 1 Li Dongfang Department of Mathematics and Statistics King Fahd University of Petroleum and Minerals P.O. Box 546 Dhahran 31261 Saudi Arabia kfupm.edu.sa 2018 132018 2018 11 10 2017 25 12 2017 132018 2018 Copyright © 2018 Jamal H. Al-Smail et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We consider the following nonlinear parabolic equation: ut-div(|u|p(x)-2u)=f(x,t), where f:Ω×(0,T)R and the exponent of nonlinearity p(·) are given functions. By using a nonlinear operator theory, we prove the existence and uniqueness of weak solutions under suitable assumptions. We also give a two-dimensional numerical example to illustrate the decay of solutions.

King Fahd University of Petroleum and Minerals FT 161004
1. Introduction

Let Ω be a bounded domain in Rn with a smooth boundary Ω. We consider the following initial and boundary value problem:Put-divupx-2u=fx,t,inΩ×0,Tux,t=0,onΩ×0,Tux,0=u0x,inΩ,where f:Ω×(0,T)R and u0:ΩR are given functions. The exponent p(·) is a given measurable function on Ω¯ such that(1)2<p1pxp2<+,with(2)p1essinfxΩpx,p2esssupxΩpx.We also assume that p(·) satisfies the log-Hölder continuity condition:(3)px-py-Alogx-y,x,yΩ,withx-y<δ,where A>0 and 0<δ<1 are constants. The term div(|u|p(·)-2u) is called the p(·)-Laplacian and denoted by Δp(·)u.

The study of partial differential equations involving variable-exponent nonlinearities has attracted the attention of researchers in recent years. The interest in studying such problems is stimulated and motivated by their applications in elastic mechanics, fluid dynamics, nonlinear elasticity, electrorheological fluids, and so forth. In particular, parabolic equations involving the p(·)-Laplacian are related to the field of image restoration and electrorheological fluids which are characterized by their ability to change the mechanical properties under the influence of the exterior electromagnetic field. The rigorous study of these physical problems has been facilitated by the development of the Lebesgue and Sobolev spaces with variable exponents.

Regarding parabolic problems with nonlinearities of variable-exponent type, many works have appeared. We note here that most of the results deal with blow-up and global nonexistence. Let us mention some of these works. For instance, Alaoui et al.  considered the following nonlinear heat equation:(4)utx,t-divumx-2u=uupx-2+f,in a bounded domain in ΩRn (n1) with a smooth boundary Ω. Under appropriate conditions on the exponent functions m,p and for f=0, they showed that any solution with nontrivial initial datum blows up in finite time. They also gave a two-dimensional numerical example to illustrate their result. Pinasco  studied the following problem:(5)ut-Δu=fu,inΩ×0,Tux,t=0,onΩ×0,Tux,0=u0x,inΩ,where ΩRn is a bounded domain with a smooth boundary Ω, and the source term is of the following form:(6)fu=axupxorfu=axΩuqyy,tdy,with p,q: Ω(1,) and the continuous function a: ΩR being given functions satisfying specific conditions. They established the local existence of positive solutions and proved that solutions with initial data sufficiently large blow up in finite time. Parabolic problems with sources like the ones in (5) appear in several branches of applied mathematics and they have been used to model chemical reactions, heat transfer, or population dynamics.

Recently, Shangerganesh et al.  studied the following fourth-order degenerate parabolic equation:(7)ut+divΔupx-2Δu=f-divg,in a bounded domain ΩRn (n1) with a smooth boundary Ω, and proved the existence and uniqueness of weak solutions of (7) by using the difference and variation methods under suitable assumptions on f,g and the exponents p.

Equation P is a nonlinear diffusion equation which has been used to study image restoration and electrorheological fluids (see ). In particular, Bendahmane et al.  proved the well-posedness of a solution, for L1-data. Akagi and Matsuura  gave the well-posedness for L2 initial datum and discussed the long-time behaviour of the solution using the subdifferential calculus approach. In our paper, we give an alternative proof of the well-posedness of  P which is simpler than that in  using a theory of nonlinear evolution equations. In addition, we give a numerical example in 2D to illustrate the decay result obtained in .

This paper consists of three sections in addition to the introduction. In Section 2, we recall the definitions of the variable-exponent Lebesgue spaces, Lp(·)(Ω), the Sobolev spaces, W1,p(·)(Ω), as well as some of their properties. We also state, without proof, a proposition to be used in the proof of our main result. In Section 3, we state and prove the well-posedness of solution to our problem. In Section 4, we give a numerical verification of the decay result.

2. Preliminaries

We present some preliminary facts about the Lebesgue and Sobolev spaces with variable exponents (see [1, 1416]). Let p:Ω[1,] be a measurable function, where Ω is a domain of Rn. We define the Lebesgue space with a variable-exponent p(·) by(8)Lp·Ωu:ΩR;measurableinΩ:ϱLp·Ωλu<,forsomeλ>0,where(9)ϱLp·Ωu=Ωuxpxdxis called a modular. Equipped with the Luxembourg-type norm,(10)uLp·Ωinfλ>0:ϱLp·Ωuλ1,Lp(·)(Ω) is a Banach space (see ).

Lemma 1 (Hölder’s inequality [<xref ref-type="bibr" rid="B15">10</xref>]).

Let p,q,s1 be measurable functions defined on Ω such that(11)1sy=1py+1qy, for a.e. yΩ. If fLp(·)(Ω) and gLq(·)(Ω), then fgLs(·)(Ω) and(12)fgs·2fp·gq·.

Lemma 2 (see [<xref ref-type="bibr" rid="B15">10</xref>]).

Let p be a measurable function on Ω. Then,

fp(·)1 if and only if ϱp(·)(f)1;

for fLp(·)(Ω), if fp(·)1, then ϱp(·)(f)fp(·); and if fp(·)1, then fp(·)ϱp(·)(f);

fp(·)1+ϱp(·)(f).

Lemma 3 (see [<xref ref-type="bibr" rid="B15">10</xref>]).

If (1) holds, then(13)minup·p1,up·p2ϱp·umaxup·p1,up·p2, for any uLp(·)(Ω).

We next define the variable-exponent Sobolev space W1,p(·)(Ω) as follows:(14)W1,p·Ω=uLp·Ωsuchthatuexists,uLp·Ω. This space is a Banach space with respect to the norm uW1,p(·)(Ω)=up(·)+up(·). Furthermore, W01,p(·)(Ω) is the closure of C0(Ω) in W1,p(·)(Ω). The dual of W01,p(·)(Ω) is defined as W-1,p(·)(Ω), by the same way as the usual Sobolev spaces where 1/p(·)+1/p(·)=1.

Lemma 4 (see [<xref ref-type="bibr" rid="B15">10</xref>]).

Let Ω be a bounded domain of Rn and p(·) satisfies (1) and (3), and then(15)up·Cup·,uW01,p·Ω,where the positive constant C depends on p(·) and Ω. In particular, the space W01,p(·)(Ω) has an equivalent norm given by uW1,p(·)(Ω)=up(·).

Lemma 5 (see [<xref ref-type="bibr" rid="B15">10</xref>]).

If p(·)C(Ω¯), q:Ω[1,) is a continuous function and(16)essinfxΩpx-qx>0withpx=npxesssupxΩn-pxifp2<nifp2n. Then the embedding W01,p(·)(Ω)Lq(·)(Ω) is continuous and compact.

Definition 6 (see [<xref ref-type="bibr" rid="B20">17</xref>]).

Let V be a separable Banach space and H be a Hilbert space such that VHV with continuous embedding and V is dense in H. Let A:VV be a nonlinear operator.

A is said to be monotone if A(u)-A(v),u-vV×V0. If, in addition, we have(17)Au-Av,u-vV×V0,uv,then A is said to be strictly monotone.

A is said to be bounded, if A(S) is bounded in V, whenever S is bounded in V.

A is said to be hemicontinuous, if the real function(18)λAu+λv,wis continuous from R to R, for any fixed u,v,wV.

We end this section with a proposition which is exactly like Theorem 7.1 .

Proposition 7.

Let u0H and fLp((0,T),V). Suppose that A: VV is a bounded monotone and hemicontinuous (nonlinear) operator satisfying, for some α,β>0 and for some p>1,(19)Av,vαvVp-β,vV.Then the following problem(20)ut+Au=fu·,0=u0,has a unique weak solution:(21)uLp0,T,VwithutLp0,T,V, where 1/p+1/p=1.

3. Well-Posedness

In this section, we state and prove the well-posedness of our problem.

Theorem 8.

Let u0L2(Ω),fLp(·)((0,T),W-1,p(·)(Ω)). Assume that (1) and (3) hold. Then P has a unique weak solution:(22)uLp10,T,W01,p·ΩL0,T,L2Ω,utLp20,T,W-1,p·Ω,where 1/p(·)+1/p(·)=1.

Proof.

We verify the conditions of Proposition 7. Let V=W01,p(·)(Ω), and equip it with the norm,(23)uW1,p·Ω=up·. So, V=W-1,p(·)(Ω). Define A:VV by(24)Au=-divupx-2u.Boundedness of  A. For all u,vV,(25)Au,v=Ωupx-2u·vΩupx-1v.Hölder inequality gives(26)Ωupx-1v2up·-1p·vV.Combining (25) and (26), we obtain(27)AuV2up·-1p·.Then, Lemma 2 implies(28)up·-1p·1+ϱp·u.Combining (27) and (28), we arrive at(29)AuV21+ϱp·u.Let SV such that uVM, for all uS. That is, up(·)M.

If M1, then Lemma 2 implies ϱp(·)(u)1 and (29) gives A(u)V4<+.

If M>1, then ϱp(·)(u)Mp2. Thus, (29) implies A(u)V2(1+Mp2)<+.

Hence A is bounded.

Monotonicity of   A . Let u,vV. Then(30)Au-Av,u-vV×V=-divupx-2u-vpx-2vu-vdx=upx-2u-vpx-2vu-vdx. By using the inequality,(31)apx-2a-bpx-2b·a-b0,for all a,bRn and a.e. xΩ. Thus we obtain A(u)-A(v),u-v0.

Hence, A is monotone.

To verify (19), we note that, for all uV, we have(32)Au,u=Ωupxdx=ϱp·u.If uV=up(·)>1, then by Lemma 3, we get(33)ϱp·uup·p1.Combining (32) and (33), we easily see that(34)Au,uuVp1.

If uV=up(·)1, then by Lemma 2, we obtain(35)Au,u=ϱp·uuV-1uVp1-1. Therefore, we have(36)Au,uuVp1-1,uV.Hemicontinuity of  A. Let u,v,wV be fixed. Let(37)gλ=Au+λv,w=Ωu+λvpx-2u+λv·wdx.Let λkλ (real) and consider(38)gλk=Ωu+λkvpx-2u+λkv·wdx. Since(39)u+λkvpx-2u+λkv·wu+λvpx-2u+λv·w, for a.e. xΩ and (40)u+λkvpx-1wCupx-1w+vpx-1wL1Ω, where C=max{2p2-2,2p2-2(1+|M|p2-1)}>0, then, by the classical dominated convergence theorem,(41)gλkgλask. Hence, A is hemicontinuous.

Therefore, conditions of Proposition 7 are satisfied and problem P has a unique solution.

4. Numerical Study

In this section, we present some numerical results and applications of the problem:Put-divupx,y-2u=0,inQ=Ω×0,Tu=0,onQ=Ω×0,Tux,y,0=u0x,y,inΩ,which is a well-posed problem due to Theorem 8. Our objective is to provide a numerical verification of the following decay result:

Proposition 9 (see [<xref ref-type="bibr" rid="B3">13</xref>]).

Assume that (1) and (3) hold. Then the solution of P satisfies the following:

If p2=2, then there exists a constant c2>0 such that (42)ut2u02e-c2t,t0.

If p2>2, there exists a constant c1>0 and t10 such that(43)ut2u021+c1t-1/p2-2,tt1.

We consider two applications to illustrate numerically an exponential decay for the case p(x,y)=2 and a polynomial decay for an exponent function p(·) satisfying conditions (1)–(3).

For this purpose, we introduce a numerical scheme for P, prove its convergence in Section 4.1, and show the decay results in Section 4.2.

4.1. Numerical Method

In this part, we present a linearized numerical scheme to obtain the numerical results of the system P and confirm the decay results. The system is fully discretized through a finite difference method for the time variable and a finite element Galerkin method for the space variable. Useful background about the numerical and error analysis of these methods is found in . More interestingly in , Li and Wang introduced a numerical scheme to solve strongly nonlinear parabolic systems and proved unconditional error estimates of the scheme. Our problem P is highly nonlinear due to the presence of the gradient and nonlinear exponent in the diffusivity coefficient, which can be zero inside the spatial domain. Below, we introduce our numerical scheme for the purpose of confirming the decay results.

The parabolic equation(44)ut-divupx,y-2u=0,inQ=Ω×0,Tis discretized using finite differences for the time derivative and a finite element method for the p(·)-Laplacian term. For this, we divide the time interval [0,T] into N equal subintervals by(45)tn=nτ,τ=TN and denote by(46)unx,yux,y,tn,n=0,1,,N.The term ut is approximated using the first-order forward finite difference formula:(47)utn+1un+1-unτ.Semidiscrete Problem. A linear semidiscrete formulation of P takes the following form: given p and u0, find {u1,u2,,un+1} such that(48)utn+1-divunpx,y-2un+1=0,inΩun+1=0,onΩu0=u0x,y,inΩ. This problem is elliptic and admits a unique solution , for every n=0,1,,N. Also, the Rothe approximation u(n) to the exact solution u, given by(49)unx,y,tui-1x,y+t-ti-1ui-ui-1τ,tti-1,ti,i=1,2,,n,is well defined and u(n)u in L2(Ω) as τ0, see .

Full-Discrete Problem. The variable un+1 is discretized in space by a finite element method. For this, let Ωh be a triangulation of Ω with a maximal element size h. Let also vh be a test function in the linear Lagrangian space P1(Ωh) such that vh=0 on Ωh.

The semidiscrete problem is then written in a weak form to define the full-discrete problem: given ph, unP1(Ωh), find uhn+1P1(Ωh) such that(50)Ωhuhn+1-uhnτvh+uhnphx,y-2uhn+1·vhdΩh=0,vhP1Ωh.For ph2, the above problem has a unique solution uhn+1H01(Ωh) for every nontrivial uhnH1(Ωh). This follows from the Lax-Milgram Lemma, and the Galerkin approximation uhn+1 converges to un+1 in H01(Ωh) as h0; see .

4.2. Numerical Results

In this subsection, we present the following numerical applications of P:

Exponential decay: for ph(x,y)=2, we show, for some c>0, that(51)g1tuhnuh0e-ct,t0.

Polynomial decay: for ph(x,y)=1/5x2+2.5, we show, for some c>0 and t0>0, that(52)g2tuhnuh01+ct-1/p2-2,tt0.Here, · denote the greatest integer function.

In both applications, we set the following parameters:(53)T=100,τ=0.1,h=0.1,Ωh=-5,5×-5,5.

Figure 1 shows the mesh used for Ωh, which involves 23702 triangles and 12052 vertices.

Mesh of the domain [-5,5]×[-5,5].

The initial condition is taken to be uh0(x,y)=e-0.5(x2+y2) and projected into P1(Ωh); see Figure 2.

Initial condition: uh0(x,y)=e-0.5(x2+y2).

The numerical results are obtained using the noncommercial software, FreeFem++ .

Application 1.

p h ( x , y ) = 2 satisfies the required conditions (1)–(3). Figure 3 shows the numerical solutions for t=5, t=10, t=20, and t=50.

Numerical solutions for Application 1.

u h 5

u h 10

u h 20

u h 50

With c=0.1, Figure 4(a) shows that uhn decays exponentially as(54)g1t=uhnuh0e-0.1t,0tT.

Exponential decay.

Solid line: y=e-0.1t; dashed: y=g1(t)

y = g 1 ( t ) / e - 0.1 t

This is also confirmed by Figure 4(b) that shows the ratio y=g1(t)/e-0.1t is less than one and remains decreasing for a large value of T.

Application 2.

The exponent function ph(x,y)=1/5x2+2.5, in Figure 5, satisfies the required conditions (1)–(3) as

p1=2.5, p2=7.5;

|p(x,y)-p(x0,y0)|=1/5x2-x02-202log(1/δ)/log|x-y| for |x-y|<δ with 0<δ<1.

p h ( x , y ) = 1 / 5 x 2 + 2.5 .

Figure 6 shows the numerical solutions for t=5, t=10, t=20, and t=50.

Numerical solutions for Application 2.

u h 5

u h 10

u h 20

u h 50

In this case, the solution uhn has a polynomial decay. With c=1, Figure 7(a) shows that(55)g2t=uhnuh01+t-2/11,0tT. This is also confirmed by Figure 7(b), which shows that the ratio y=g2(t)/(1+t)-2/11 remains less than one and decreasing until T.

Polynomial decay.

Solid line: y=(1+t)-2/11; dashed: y=g2(t)

y = g 2 ( t ) / ( 1 + t ) - 2 / 11

We conclude that the numerical results in above applications verify Proposition 9.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors acknowledge King Fahd University of Petroleum and Minerals for its support. This work is sponsored by KFUPM under Project no. FT 161004.

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