# Existence of Positive Solutions and Asymptotic Behavior for Evolutionaryq(x)-Laplacian Equations

Authors: Aboubacar Marcos; Ambroise Soglo
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

In this paper, we extend the variational method of M. Agueh to a large class of parabolic equations involvingq(x)-Laplacian parabolic equation ∂ρt,x/∂t=divxρt,x∇xG′ρ+Vqx−2∇xG′ρ+V. The potential V is not necessarily smooth but belongs to a Sobolev space W1,∞Ω. Given the initial datum ρ0 as a probability density on Ω, we use a descent algorithm in the probability space to discretize the q(x)-Laplacian parabolic equation in time. Then, we use compact embedding W1,q.Ω↪↪Lq.Ω established by Fan and Zhao to study the convergence of our algorithm to a weak solution of the q(x)-Laplacian parabolic equation. Finally, we establish the convergence of solutions of the q(x)-Laplacian parabolic equation to equilibrium in the p(.)-variable exponent Wasserstein space.


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## Body

## 1. Introduction and the Main Results

In this paper, we study the existence of positive solutions and the asymptotic behavior for a class of parabolic equations involving parabolic equations governed by theq (x)-Laplacian operator:(1)∂ρt,x∂t=divxρt,x∇xG′ρ+Vqx−2∇xG′ρ+V,in0,+∞×Ω,where Ω⊂ℝd is a convex, bounded, and smooth domain of ℝd, G:0,+∞⟶ℝ is a convex function of class C2]0,∞[, q.:Ω⟶1,+∞ is a continuous function, and V:Ω⟶ℝ belongs to a Sobolev space W1,∞Ω.q(x)-Laplacian parabolic equation type is a broad family of parabolic equations including many equations emerging in the mathematical modeling of a variety of phenomena in physics such as the flow of compressible fluids in nonhomogeneous isotropic porous media, the behavior of electrorheological fluids [1, 2], image processing [3], and the curl systems emanating from electromagnetism [4, 5].Some authors have studied the existence of solutions of theq(x)-Laplacian parabolic equation with the variable exponent, when Gt=t2 and V≡0 (see [1, 2, 6]), for a given initial datum ρ0 and a homogeneous boundary condition. In their works, they use an approximation method to approach the q(x)-Laplacian parabolic equation by regularized problems under the following conditions: ρ0∈L∞Ω and ∇xρ0q+∈L1Ω.In [7], M. Agueh studied the existence of positive solutions for the q(x)-Laplacian parabolic equation when the variable exponent qx≡q is constant (with q>1), and the potential V is a convex function of class C1Ω. Moreover, the author in [7] proved that the parabolic q-Laplacian equation(2)∂ρt,x∂t=divxρt,x∇xG′ρ+Vq−2∇xG′ρ+V,in0,+∞×Ω,is a gradient flow of the functional Eρ=∫ΩGρ+ρVdx with respect to the p-Wasserstein distance (p=q/q−1) defined by(3)Wpρ1,ρ2=infS#ρ1=ρ2∫Ωx−Sxpρ1dx1/p,where ρ1 and ρ2 are two probability densities on Ω.In fact, by fixing the time steph>0 and a probability density ρ0 on Ω, the author defines ρk−1 (k∈ℕ∗) as the solution of (2) at t=hk−1 and ρk as the solution of (2) at t=hk such that ρk is the unique solution of the variational problem(4)Pk:infρ∈PΩEρ+1php−1Wppρ,ρk−1.Then, the author established the convergence of the approximate solutionsρkk to a weak solution of the Laplacian parabolic equation . Here, we extend the work of [7] to a general case where V≠0 may not be smooth but belongs to a Sobolev space W1,∞Ω. Roughly speaking, we use mass the transportation method borrowing ideas from [7, 8] to establish the existence of positive solutions and the long time behavior of solutions of the Laplacian parabolic equation. As in [7], we prove that the Laplacian parabolic equation is a gradient flow of the functional Eρ=∫ΩGρ+ρVdx with respect to the p.−Wasserstein distance Wp. defined by(5)Wp.ρ1,ρ2=infρt,.vLρt,.p.0,1×Ω,∂ρt,x∂t+divxρt,xvt,x=0,inΩ,ρ0,.=ρ1andρ1,.=ρ2.Next, we proceed with the discretization of theq(x)-Laplacian parabolic equation as follows: fixing the time step h>0 and a probability density ρ0 on Ω, we define ρk as the approximate solution of the q(x)-Laplacian parabolic equation at tk=hk, which minimizes the variational problem(6)Pk:infρ∈PΩ∫ΩGρ+ρVdx+Kp.hρk−1,ρ,where(7)Kp.hρk−1,ρ=infγ∈Πρk−1,ρ∫Ω×Ωx−ypxpxhpx−1dγ.Here,Πρk−1,ρ is the set of all probability measures on Ω×Ω having ρk−1 and ρ as their marginals, and(8)Kp.hρk−1,ρ,k≥1,is the Monge–Kantorovich work associated to the cost chx,y=x−ypx/pxhpx−1.The establishment of our result will be derived according to the following steps:(1)
Givenρ0 as a probability density on Ω such as N≤ρ0≤M and V∈L∞Ω, we prove that Pk admits a unique solution ρk which satisfies N≤ρ0≤M (see Lemma 1).(2)
We prove in (28) that the Kantorovich problem(9)K:infγ∈Πρk−1,ρk∫Ω×Ωx−ypxpxhpx−1dγ,admits a unique solutionγk in Πρk−1,ρk and that suppγk satisfy(10)suppγk⊂x,y∈Ω×Ω,y=x−h∇xG′ρk+Vqx−2∇xG′ρk+V,whereqx=px/px−1.Here,V∈W1,∞Ω being not necessarily smooth, we approximate V by C2Ω-functions, and we use descent algorithm (6), the maximum principle N≤ρk≤M, and compact embedding W1,qxΩ↪↪LqxΩ to establish (10).(3)
We now use the maximum principleN≤ρk≤M and (10) to prove that the sequence ρkk is a time discretization of the nonlinear q(x)-Laplacian parabolic equation, that is, for all test functions, ϕ∈Cc∞ℝd,(11)∫Ωρk−ρk−1ϕdx+h∫Ω∇xG′ρk+Vqx−2∇xG′ρk+V,∇xϕ>ρkdx=Akhϕ,whereAkhϕ converges to 0 when h tends to 0.(4)
We defineρht,x by(12)ρht,x=ρk,ift∈hk−1,hk,ρ0,x=ρ0.Afterward, we establish the following:(i)
The strong convergence of the sequenceρhh to a function ρt,x in Lqx0,T×Ω for 0<T<∞(ii)
The weak convergence of nonlinear termdivxρh∇xG′ρh+Vqx−2∇xG′ρh+Vh to divxρ∇xG′ρh+Vqx−2∇xG′ρ+V in Cc∞ℝ×ℝd′To prove (i), we use descent algorithm (6) and the maximum principle N≤ρh≤M to deduce that the sequence ρhh is bounded in W1,qx0,T×Ω for 0<T<∞. Then, taking into account the compact embedding W1,qx0,T×Ω↪↪ Lqx0,T×Ω established by Fan and Zhao in [9], we conclude that the sequence ρhh converges strongly to ρt,x in Lqx0,T×Ω.We now use (i) and the fact thatv⟼vqx is convex for all x∈Ω fixed to establish (ii). We combine (i) and (ii) to prove that the sequence ρhh converges to a weak solution of the q(x)-Laplacian parabolic equation.Finally, we use the energy method to study the convergence of solutions of theq(x)-Laplacian parabolic equation to ρ∞ when t⟶∞, where ρ∞∈PΩ is the probability density which satisfies ρ∞∇xG′ρ∞+V=0.Note that in [10, 11], the authors proved a convergence of solutions to the equilibrium without specifying the speed of convergence. In [11], the long-time behavior of solutions of the q(x)-Laplacian parabolic equation is only established if 2≤q−≤q+.In this paper, we extend to the variable exponentq.:Ω⟶]1,+∞[ such that 1<q−<q+<∞, the results obtained by the previous authors, and we also specify the rates of convergence.Our results in this work are stated as follows:Theorem 1.
Assume thatHρ0, HG, Hq.p., and HV1 hold, and define ρht,x:0,+∞×Ω⟶0,+∞ by(13)ρht,x=ρkx,ift∈hk−1;hk,ρ0,x=ρ0x.Then, the sequenceρhh converges to a weak solution ρt,x of the q(x)-Laplacian parabolic equation.Theorem 2.
Assume thatHρ0, HG, Hq.p., HV1, and HV2 hold. Let ρt,x be a weak solution of the q(x)-Laplacian parabolic equation. Define(14)I1≔t∈0,+∞,∇xG′ρt,.+VLρt,.q.Ω≥1,I2≔t∈0,+∞,∇xG′ρt,.+VLρt,.q.Ω≤1.Then,(15)Eρt2,.−Eρ∞≤Eρt,.−Eρ∞1+λ/2p+′q+q+−1Eρt2,.−Eρ∞q+−1t−t21/q+−1,onI2,Wp.p+ρt,.,ρ∞.≤2p+Eρt2,.−Eρ∞λ1+λ/2p+′q+q+−1Eρt2,.−Eρ∞q+−1t−t21/q+−1,onI2,Eρt,.−Eρ∞≤e−λ/2p+′t−t1Eρ0−Eρ∞,onI1,(16)Wp.p+ρt,.,ρ∞.≤2p+λe−λ/2p+′t−t1Eρ0−Eρ∞,onI1,where t1≔infI1 and t2≔infI2.


## 2. Preliminaries

### 2.1. Assumptions

Throughout this work, we will assume the following:Hρ0ρ0 is a probability density on Ω, with N≤ρ0≤M, for some 0<N<M, and ∫ΩGρ0dx<∞.Hq.p.: q.,p.:Ω⟶]1,+∞[ are continuous functions such that 1/px+1/qx=1 for all x∈Ω, and 1<p≔minΩpx≤maxΩpx≔p+; 1<q≔minΩqx≤maxΩqx≔q+.HGG:0,+∞⟶ℝ is strictly convex, G∈C2]0,+∞[, and t⟼tdGt−d is convex.(17)HV1V∈W1,∞Ω.HV2V:ℝd⟶ℝ is a potential which satisfies V0=0, V∈C2Ω, and(18)Vx−Vy≥∇Vy.x−y+λx−yp+,for allx,y∈Ω,for someλ>0.


### 2.2. Lebesgue–Sobolev Spaces with Variable Exponents

We recall in this section some definitions and fundamental properties of the Lebesgue and Sobolev space with variable exponents.Definition 1.
Letρ be a probability measure on Ω and p.:Ω⟶]1,+∞[ be a measurable function. We denote by Lρp.Ω the Lebesgue space with the variable exponent defined by(19)Lρp.Ω≔u:Ω⟶ℝ;∫Ωuxλpxρxdx≤1,for someλ>0,with the norm(20)uLρp.Ω=infλ>0,∫Ωuxλpxρxdx≤1,for all u∈Lρp.Ω.
We denote byWρ1,p.Ω the Sobolev space with the variable exponent defined by(21)Wρ1,p.Ω≔u∈Lρp.Ω,∇u∈Lρp.Ω,equipped with the norm(22)uWρ1,p.Ω≔uLρp.Ω+∇uLρp.Ω.
In particular, ifρx=1, then Lρp.Ω=Lp.Ω and Wρ1,p.Ω=W1,p.Ω.
We also define the spaceW01,p.Ω as the closure of the space of C∞-functions with compact support in ΩCc∞Ω in the space W1,p.Ω endowed with the norm(23)uW01,p.Ω≔∇uW1,p.Ω.
It is well known (cf. [9]) that Lρp.Ω and Wρ1,p.Ω are Banach spaces, respectively, with norms (20) and (22).Proposition 1 (Hölder inequality; see [12]).
Letρ∈PΩ and p.,q.:Ω⟶]1,+∞[ be two measurable functions such that 1/px+1/qx=1, for all x∈Ω.
Ifu∈Lρp.Ω and v∈Lρq.Ω, then(24)∫Ωuxvxρxdx≤1p−+1q−uLρp.ΩvLρq.Ω.Furthermore, ifp1.,p2.,p3.:Ω⟶]1,+∞[ are measurable functions such that 1/p1x=1/p2x+1/p3x, for almost all x∈Ω, we have(25)uvLρp1.Ω≤2uLρp2.ΩvLρp3.Ω,for u∈Lρp2.Ω and v∈Lρp3.Ω.Proposition 2 (see [9]).
Letρ∈PΩ and p1.,p2.:Ω⟶1,+∞ be two measurable functions such that p1x≤p2x on Ω. Then, we have the following continuous injection:(26)Lρp2.Ω⟼Lρp1.Ω.Furthermore,(27)uLρp1.Ω≤2uLρp2.Ω,for all u∈Lρp2.Ω.Proposition 3 (see [9]).
Assume thatHq.p. holds. Then, the following statements hold:(i)
The Banach spacesWρ1,pxΩ and Lρp.Ω are separable and reflexive Banach spaces(ii)
The embeddingW1,qxΩ↪↪LqxΩ is continuous and compact(iii)
There is a constantCΩ such that uW1,qxΩ≤CΩ∇uLqxΩ for all u∈W01,qxΩ


## 2.1. Assumptions

Throughout this work, we will assume the following:Hρ0ρ0 is a probability density on Ω, with N≤ρ0≤M, for some 0<N<M, and ∫ΩGρ0dx<∞.Hq.p.: q.,p.:Ω⟶]1,+∞[ are continuous functions such that 1/px+1/qx=1 for all x∈Ω, and 1<p≔minΩpx≤maxΩpx≔p+; 1<q≔minΩqx≤maxΩqx≔q+.HGG:0,+∞⟶ℝ is strictly convex, G∈C2]0,+∞[, and t⟼tdGt−d is convex.(17)HV1V∈W1,∞Ω.HV2V:ℝd⟶ℝ is a potential which satisfies V0=0, V∈C2Ω, and(18)Vx−Vy≥∇Vy.x−y+λx−yp+,for allx,y∈Ω,for someλ>0.


## 2.2. Lebesgue–Sobolev Spaces with Variable Exponents

We recall in this section some definitions and fundamental properties of the Lebesgue and Sobolev space with variable exponents.Definition 1.
Letρ be a probability measure on Ω and p.:Ω⟶]1,+∞[ be a measurable function. We denote by Lρp.Ω the Lebesgue space with the variable exponent defined by(19)Lρp.Ω≔u:Ω⟶ℝ;∫Ωuxλpxρxdx≤1,for someλ>0,with the norm(20)uLρp.Ω=infλ>0,∫Ωuxλpxρxdx≤1,for all u∈Lρp.Ω.
We denote byWρ1,p.Ω the Sobolev space with the variable exponent defined by(21)Wρ1,p.Ω≔u∈Lρp.Ω,∇u∈Lρp.Ω,equipped with the norm(22)uWρ1,p.Ω≔uLρp.Ω+∇uLρp.Ω.
In particular, ifρx=1, then Lρp.Ω=Lp.Ω and Wρ1,p.Ω=W1,p.Ω.
We also define the spaceW01,p.Ω as the closure of the space of C∞-functions with compact support in ΩCc∞Ω in the space W1,p.Ω endowed with the norm(23)uW01,p.Ω≔∇uW1,p.Ω.
It is well known (cf. [9]) that Lρp.Ω and Wρ1,p.Ω are Banach spaces, respectively, with norms (20) and (22).Proposition 1 (Hölder inequality; see [12]).
Letρ∈PΩ and p.,q.:Ω⟶]1,+∞[ be two measurable functions such that 1/px+1/qx=1, for all x∈Ω.
Ifu∈Lρp.Ω and v∈Lρq.Ω, then(24)∫Ωuxvxρxdx≤1p−+1q−uLρp.ΩvLρq.Ω.Furthermore, ifp1.,p2.,p3.:Ω⟶]1,+∞[ are measurable functions such that 1/p1x=1/p2x+1/p3x, for almost all x∈Ω, we have(25)uvLρp1.Ω≤2uLρp2.ΩvLρp3.Ω,for u∈Lρp2.Ω and v∈Lρp3.Ω.Proposition 2 (see [9]).
Letρ∈PΩ and p1.,p2.:Ω⟶1,+∞ be two measurable functions such that p1x≤p2x on Ω. Then, we have the following continuous injection:(26)Lρp2.Ω⟼Lρp1.Ω.Furthermore,(27)uLρp1.Ω≤2uLρp2.Ω,for all u∈Lρp2.Ω.Proposition 3 (see [9]).
Assume thatHq.p. holds. Then, the following statements hold:(i)
The Banach spacesWρ1,pxΩ and Lρp.Ω are separable and reflexive Banach spaces(ii)
The embeddingW1,qxΩ↪↪LqxΩ is continuous and compact(iii)
There is a constantCΩ such that uW1,qxΩ≤CΩ∇uLqxΩ for all u∈W01,qxΩ


## 3. Existence of Solutions for the Nonlinearq(x)-Laplacian Parabolic Equation

In this section, we prove the existence of solutions for the nonlinearq(x)-Laplacian parabolic equation.


### 3.1. Euler–Lagrangian Equation for Variational Problem (37)

Here, we show that the sequenceρkk defined in (37) is a time discretization of the q(x)-Laplacian parabolic equation, i.e., for all test functions, ϕ∈Cc∞ℝd,ℝ, we have(28)∫Ωρk−ρk−1ϕdx+h∫Ω<∇xG′ρk+Vqx−2∇xG′ρk+V,∇xϕ>ρkdx=Akhϕ,where Akhϕ converges to 0 when h tends to 0.Lemma 1.
Assume thatHρ0, HG, and HV1 hold. Then, the variational problem(29)P1:infρ∈PΩIρ≔∫ΩGρ+ρVdx+Kp.hρ0,ρ,where(30)Kp.hρ0,ρ≔infγ∈Πρ0,ρ∫Ω×Ωx−ypxpxhpx−1dγ,admits a unique solution ρ1 in PΩ which satisfies N≤ρ1≤M.Proof 1.
SinceN≤ρ0≤M, then the solution ρ1 of variational problem P1 (if there exists) satisfies N≤ρ1≤M. The proof of the maximum principle is carried out similarly as given in [7].
Since∫ΩGρ0dx<∞ and V∈L∞Ω, we have Iρ0<∞.
Letρ∈PΩ; since G is convex and V∈L∞Ω,(31)Iρ≥∫ΩGρ+ρVdx≥ΩG1Ω−VL∞Ω.
We conclude thatinfρ∈PΩIρ is finite.
Letρnn be a sequence in PΩ such that N≤ρn≤M and Iρn converges to infρ∈PΩIρ. Then, ρnn converges weakly∗ to ρ1 in L∞Ω up to a subsequence.
G being positive and convex, then(32)∫ΩGρn+ρnVdx≥∫ΩGρ1+ρ1Vdx+∫ΩG′ρ1ρn−ρ1dx+∫Ωρn−ρ1Vdx.
Next, we use the fact thatG′ρ1∈L∞Ω and V∈L∞Ω to obtain(33)liminfn⟶∞∫ΩGρn+ρnVdx≥∫ΩGρ1+ρ1Vdx.
Since the variable exponentp. is continuous on Ω, the Kantorovich problem(34)K:infγ∈Πρ0,ρn∫Ω×Ωx−ypxpxhpx−1dγ,admits a solution γn. Moreover, since Ω is bounded, the sequence γnn converges to a measure γ1 in PΩ×Ω narrowly, and γ1∈Πρ0,ρ1; and then, we derive that(35)liminfn⟶∞Kp.hρ0,ρn≥∫Ω×Ωx−ypxpxhpx−1dγ1≥Kp.hρ0,ρ1.
We combine (35) and (33) to obtain(36)liminfn⟶∞Iρn≥Iρ1.
Thus,ρ1 is a solution of variational problem P1. From the strict convexity of G, we deduce that ρ⟼∫ΩGρ+ρVdx is strictly convex and so is ρ⟼Iρ on PΩ, and consequently, the uniqueness of the solution ρ1 of P1 follows.Now, we assume thatHρ0, HG, and HV1 hold. Then, from Lemma 1, we obtain that the variational problem(37)Pk:infρ∈PΩIρ≔∫ΩGρ+ρVdx+Kp.hρk−1,ρ,with(38)Kp.hρk−1,ρ≔infγ∈Πρk−1,ρ∫Ω×Ωx−ypxpxhpx−1dγ,admits a unique solution ρk for all k≥1.Next, we prove thatρkk is a time discretization of the nonlinear q(x)-Laplacian parabolic equation. In order to achieve this, we use the following lemma.Lemma 2.
Assume thatHρ0, HG, and HV1 hold. Then, the Kantorovich problem(39)K:infγ∈Πρk−1,ρk∫Ω×Ωx−ypxpxhpx−1dγ,admits a solution γk such that(40)suppγk⊂x,y∈Ω×Ω,y=x−h∇xG′ρk+Vqx−2∇xG′ρk+V,where qx=px/px−1.Proof 2.
The proof of Lemma2 is derived following the two steps.Step 1.
We first assume thatV∈C2Ω.
Fixψ∈Cc∞ℝd,ℝd. Let Tϵϵ∈ℝ in C∞ℝd,ℝd defined by(41)Tϵx=x+ϵψx.
Define the probability densityρϵ as ρϵ=Tϵ#ρk and γϵ as a probability measure on Ω×Ω defined by(42)∫Ω×Ωϕx,ydγϵ=∫Ω×Ωϕx,Tϵydγk,for all ϕ∈Cb0Ω×Ω.
Forϵ≠0,(43)Iρϵ−Iρkϵ=∫ΩGρϵ−Gρkϵdx+∫ΩVTϵ−Vxϵdx+1ϵKp.hρϵ,ρk−1−Kp.hρk−1,ρk.
SinceG satisfies HG, then(44)limϵ⟶0∫ΩGρϵ−Gρkϵdx=∫Ω∇xG′ρk.ψρkdx.
SinceV∈C2Ω, from the Taylor formula, we have (see [7])(45)VTϵx−Vx=ϵ∇xVx⋅ψx+ϵ2ψxτ∇x2Vx+ϵθψx⋅ψx,withθ∈0,1.
Then, we have after integration,(46)∫ΩVTϵ−Vxϵdx=∫Ω<∇Vx,ψ>ρkdx+ϵ∫Ωψxτ⋅∇2Vx+ϵθψx⋅ψxρkdx.
By using (41) and the fact that V∈W1,∞Ω, we have(47)ψxτ.∇2Vx+ϵθψx.ψx≤∇2VL∞ΩψL∞Ω.
We now use the dominated convergence theorem to have(48)limϵ⟶0∫ΩVTϵ−Vxϵdx=∫Ω∇V.ψρkdx.
Note thatγϵ defined in (42) belongs to Πρk−1,ρϵ, and(49)Kp.hρk−1,ρk=∫Ω×Ωx−ypxpxhpx−1dγk,Kp.hρk−1,ρϵ≤∫Ω×Ωx−Tϵypxpxhpx−1dγk.
So, forϵ>0, we have(50)1ϵKp.hρk−1,ρϵ−Kp.hρk−1,ρk≤1ϵ∫Ω×Ωx−Tϵypx−x−ypxpxhpx−1dγk.
Note that(51)limϵ⟶0+1ϵ∫Ω×Ωx−Tϵy|px−|x−y|pxpxhpx−1dγk=−∫Ω×Ω<|x−yh|px−2x−yh,ψ>dγk.
Indeed,(i)
We have(52)limϵ⟶0+1ϵx−Tϵypx−x−ypxpxhpx−1=−x−yhpx−2x−yh.ψ.(ii)
On the contrary, the Taylor formula with respect toϵ enables us to write(53)x−Tϵypx−x−ypxpxhpx−1=−ϵx−y−θϵψyhpx−2x−y−θϵψyh.ψy,where θ∈0,1 and Tϵy=y+ϵψy. Also, we have(54)x−Tϵypx−x−ypxϵpxhpx−1=x−y−θϵψyhpx−1ψy≤x−y−θϵψyh+1px−1ψy≤x−y−θϵψyh+1pxψy≤2p+−1x−yhpx+2p+−1ψyh+1p+ψy,with(55)∫Ω×Ω2p+−1x−yhpx+2p+−1ψyh+1p+ψydγkx,y<∞.
We then combine the results given in (i) and (ii) and the dominated convergence theorem to obtain (50).
Sinceρk minimizes I on PΩ, then(56)limϵ⟶0Iρϵ−Iρkϵ=0.
So, we combine (44), (48), and (50) to obtain that(57)0=limϵ⟶0+Iρϵ−Iρkϵ≤∫Ω∇xG′ρk+V.ψρkdx−∫Ω×Ω<x−yhpx−2x−yh,ψ>dγk.
Replacingψ by −ψ in (56), we get(58)∫Ω<∇xG′ρk+V,ψ>ρkdx−∫Ω×Ω<x−yhpx−2x−yh,ψ>dγk≤0.
From (56) and (57), we deduce that(59)∫Ω<∇xG′ρk+V,ψ>ρkdx+∫Ω×Ω<y−xhpx−2y−xh,ψ>dγk=0.
Thus, we obtain (40) when V∈C2Ω.
Now, let us establish the proof of the lemma whenV∈W1,∞Ω and is nonregular.Step 2.
Assume thatV∈W1,∞Ω. Let Vδδ be a sequence in C2Ω such that limδ⟶0Vδ−VW1,∞Ω=0.
By fixingδ>0, we define the sequence ρkδk such that ρ0δ=ρ0 with ρkδ (for k≥1), the solution of the variational problem(60)Pkδ:infρ∈PΩ∫ΩGρ+ρVδdx+Kp.hρk−1δ,ρ,where Kp.hρk−1δ,ρ is defined as in (38). As in Lemma 1, the variational problem Pkδ admits a unique solution ρkδ in PΩ, and N≤ρkδ≤M. Hence, the Kantorovich problem(61)Kδ:infγ∈Πρk−1δ,ρkδ∫Ω×Ωx−ypxpxhpx−1dγ,admits a solution γkδ such that(62)suppγkδ⊂x,y∈Ω×Ω,y=x−h∇xG′ρkδ+Vδqx−2∇xG′ρkδ+Vδ.
Let us show thatρkδδ converges to ρk and γkδδ converges to γk up to a subsequence, as well as suppγk satisfies (40)
UsingPkδ and (61), we have(63)∫ΩGρk−1δ+ρk−1δVδdx−∫ΩGρkδ+ρkδVδdx≥h∫Ω∇xG′ρkδ+Vδqxpxρkδdx.
However,G is convex, and V∈W1,∞Ω; and then,(64)∫Ω∇xG′ρkδ+Vδqxρkδ≤p+h∫ΩGρ0dx−ΩG1Ω+2VL∞Ω.
Furthermore, recalling (63) and the fact that N≤ρkδ≤M, ∇xG′ρkδ=G′′ρkδ∇xρkδ, and ∇xG′ρkδqx≤2q+−1∇xG′ρkδ+Vδqx+∇Vδqx, we obtain(65)∫Ω∇xρkδqxdx≤2q+−1k0p+Nh∫ΩGρ0dx−ΩG1Ω+2VL∞Ω+∇VL∞Ωq+,where k0=inft,x∈N,M×ΩG′′tqx. We conclude that ρkδδ is bounded on W1,qxΩ. Since q. is continuous and Ω is a bounded set, the injection W1,qxΩ↪↪LqxΩ is compact (see [9]), and hence, the sequence ρkδδ converges strongly to some uk in LqxΩ up to a subsequence. Moreover, ρkδ minimizes Pkδ for k∈ℕ∗, and then, we have(66)∫ΩGρ+ρVδdx+Kp.hρk−1δ,ρ≥∫ΩGρkδ+ρkδVδdx+Kp.hρk−1δ,ρkδ,for all ρ∈PΩ. Also, from the boundedness of Ω, γkδδ converges narrowly to a measure Γk in PΩ×Ω, and Γk∈Πuk−1,uk. Then, we use the strong convergence of ρkδδ to uk and the fact that Vδ−VW1,∞Ω converges to 0 when δ⟶0 to obtain that(67)∫ΩGρ+ρVdx+Kp.huk−1,ρ≥∫ΩGuk+ukVdx+Kp.huk−1,uk,for all ρ∈PΩ. Since ρ0δ=ρ0, we conclude that uk=ρk and Γk=γk for all k∈ℕ∗.
By using (63) and N≤ρk≤M, we deduce that ∇xG′ρkδ+Vδδ is bounded in LqxΩd. Since ρkδδ converges strongly to ρk in LqxΩ and G′ is continuous, then ∇xG′ρkδ+Vδδ converges weakly to ∇xG′ρk+V in LqxΩd.
Next, we use (57), and we have(68)∫Ω<∇xG′ρkδ+Vδ,ψ>ρkδdx+∫Ω×Ω<y−xhpx−2y−xh,ψ> dγkδ=0.
Sinceρkδδ converges strongly to ρk in LqxΩ, ∇xG′ρkδ+Vδδ converges weakly to ∇xG′ρk+V in LqxΩd, and γkδδ converges narrowly to γk in PΩ×Ω; then, tending δ to 0 in (67), we obtain(69)∫Ω<∇xG′ρk+Vδ,ψ>ρkdx+∫Ω×Ω<y−xhpx−2y−xh,ψ> dγk=0,for all test functions, ψ∈Cc∞ℝd,ℝd.
Finally, we obtain the equality(70)y=x−h∇xG′ρk+Vqx−2∇xG′ρk+Vγka.e.
Now, let us prove thatρkk is a time discretization of a nonlinear q(x)-Laplacian parabolic equation. Let ψ∈Cc∞ℝd,ℝ be a test function; we have(71)∫Ωρk−ρk−1ψdx=∫Ω×Ωψy−ψxdγkx,y.
We now use (69) and the Taylor formula to obtain(72)∫Ωρk−ρk−1ψdx=−h∫Ω<∇xG′ρk+Vqx−2∇xG′ρk+V,∇ψ>ρkdx+h2∫Ω×Ω<∇xG′ρk+Vτ,∇2ψx+θy−x∇xG′ρk+V>dγkx,y,where θ∈0,1 and ∇xG′ρk+Vτ is the transpose of ∇xG′ρk+Vτ.
Let us show that(73)Akhψ=h2∫Ω×Ω<∇xG′ρk+Vτ,∇2ψx+θy−x∇xG′ρk+V>dγkx,yconverge to 0 when h tends to 0. Since qx−1=qx/px, we have(74)Akhψ≤h2∇2ψL∞Ω∫Ω∇xG′ρk+V2qx/pxρkdx.
DefineΩ1≔x∈Ω,px≥2 and Ω1≔x∈Ω,px<2:(75)∫Ω1∇xG′ρk+V2qx/pxρkdx≤2q+−1∫Ω∇xG′ρk+Vqxρkdx+1,∫Ω2∇xG′ρk+V2qx/pxρkdx≤h1−2p−∫Ω∇xG′ρk+Vqxρkdx.
We use (64) and (73) in (72), and we have(76)Akhψ≤2p+−1k0h2∇xψL∞Ω+p+h∇xψL∞Ω+h2−2/p−∫ΩGρ0dx−ΩG1Ω+2∇L∞Ω.
We tendh to 0, and we conclude that Akhψ converges to 0. Thus, the sequence ρkk is a time discretization of the nonlinear q(x)-Laplacian parabolic equation.
Defineρht,x:0,+∞×Ω⟶0,+∞ as follows:(77)ρht,x=ρkxift∈k−1h,kh,andρh0,x=ρ0x.
We prove in the next section that the sequenceρhh converges strongly to a function ρt,x in Lqx0,T×Ω for 0<T<∞ and that the nonlinear term sequence divxρh∇xG′ρh+Vqx−2∇xG′ρh+Vh converges weakly to divxρh∇xG′ρh+Vqx−2∇xG′ρh+Vh in ψ∈Cc∞ℝd,ℝd.


### 3.2. Strong Convergence ofρhh and Weak Convergence of the Nonlinear Term Sequence

Here, we denote the nonlinear term sequence bydivxρhωhh, with ωh≔|∇xG′ρh+Vqx−2∇xG′ρh+V.SinceN≤ρk≤M, then N≤ρh≤M. Thus, we use (64), and we have(78)∫0,T×Ω∇xρhqxdx≤2q+−1k0p+N∫ΩGρ0dx−ΩG1Ω+2VL∞Ω+∇VL∞Ωq+,where 0<T<∞. We conclude that ρht,.h is bounded in W1,qxΩ for almost all t∈0,T. By using the compact injection W1,qxΩ↪↪LqxΩ, we deduce that ρht,.h converges strongly to a function ρt,. in LqxΩ. Since N≤ρh≤M and 0,T×Ω is bounded, we use the dominated convergence theorem to have(79)limh⟶0∫0T∫Ωρht,x−ρt,xqxdxdt=0.Then, up to a subsequence, the sequenceρhh converges strongly to ρt,x in Lqx0,T×Ω. By using (64) and N≤ρh≤M, we deduce that ∇xG′ρh+Vh is bounded in Lqx0,T×Ωd. Then, since G′ is continuous and ρhh converges strongly to ρ in Lqx0,T×Ω, we conclude that ∇xG′ρh+Vh converges weakly to ∇xG′ρ+V in Lqx0,T×Ωd.Note thatωhpx=∇xG′ρh+Vqx. Then, using the fact that ∇xG′ρh+Vh is bounded in Lqx0,T×Ω, we conclude that ωhh is bounded in Lqx0,T×Ω. Thus, the sequence ωhh converges weakly to some ω∈Lpx0,T×Ω.As in [7], we derive easily that divxρh∇xG′ρh+Vqx−2∇xG′ρh+Vh converges weakly to divxρ∇xG′ρ+Vqx−2∇xG′ρ+V in ψ∈Cc∞ℝd,ℝd.Lemma 3.
FixT>0. Let u∈Cc∞ℝ be a test function such that suppu⊂−T,T. We have(80)limh⟶0∫0,T×Ω<ρhωh,∇xG′ρh+V>utdtdx=∫0,T×Ω<ρω,∇xG′ρh+V>utdtdx.Proof 3.
The proof of this lemma is achieved via the following three claims:
Claim 1:(81)liminfh⟶0∫0,T×Ωρh<ωh,∇xG′ρh+V>utdtdx≥∫0,T×Ωρ<ω,∇xG′ρh+V>utdtdx.Proof 4.
Sinceρh≥0, u≥0, and v⟼vqx−2v is monotone for all x fixed in Ω, we have(82)∫0,T×Ωρh<ωh−∇xG′ρ+Vqx−2∇xG′ρ+V,∇xG′ρh+V−∇xG′ρ+V>utdtdx≥0.
Using (81), we have(83)∫0,T×Ω<ρhωh,∇xG′ρh+V>utdtdx≥∫0,T×Ω<ρhωh,∇xG′ρ+V>utdtdx+∫0,T×Ω<ρh∇xG′ρ+Vqx−2∇xG′ρ+V,∇xG′ρh+V−∇xG′ρ+V>utdtdx.
Sinceρhh converges strongly to ρ in Lqx0,T×Ω, ∇xG′ρh+Vh converges weakly to ∇xG′ρ+V in Lqx0,T×Ω, and ωhh converges weakly to ω in Lqx0,T×Ω, then(84)liminfh⟶0∫0,T×Ω<ρhωh,∇xG′ρ+V>utdtdx=∫0,T×Ω<ρω,∇xG′ρ+V>utdtdx,(85)liminfh⟶0∫0,T×Ω<ρh∇xG′ρ+Vqx−2∇xG′ρ+V,∇xG′ρh+V−∇xG′ρ+V>utdtdx=0.
We now use (83) and (84) in (82) to conclude the proof of claim 1.
Claim 2:(86)limsuph⟶0∫0,T×Ω<ρhωh,∇xG′ρh+V>utdtdx≤∫Ωρ0G′ρ0−G∗G′ρ0u0dx+∫0,T×ΩρG′ρ−G∗G′ρhu′tdtdx+∫0,T×ΩρVu′tdtdx+∫Ωρ0Vu0dx.Proof 5.
We use the descent algorithmPk and (70) to obtain(87)∫ΩGρk−1+ρk−1Vdx−∫ΩGρk+ρkVdx≥h∫Ω∇xG′ρk+Vqxpxρkdx.
Note that<ωk,∇xG′ρk+V>=∇xG′ρk+Vqx, with ωk=∇xG′ρk+Vqx−2∇xG′ρk+V. Then, we obtain after integration over 0,T that(88)1h∑k=1T/h∫hk−1hk∫ΩGρk−1+ρk−1Vdx−∫ΩGρk+ρkVdxutdt≥∫0T∫Ω<ωk,∇xG′ρk+V>pxρkutdtdx.
We usepx≤p+ and τ=h/p+ in (87), and we have(89)1h∑k=1T/h∫hk−1hk∫ΩGρk−1+ρk−1Vdx−∫ΩGρk+ρkVdxutdt≥∫0T∫Ω<ωτ,∇xG′ρτ+V>ρτutdtdx,where ρτ is defined as follows:(90)ρτt,x=ρkx,ift∈τk−1,τk,ρτ0,x=ρ0x,and ωτ=∇xG′ρτ+Vqx−2∇xG′ρτ+V.
Using the fact thatsuppu⊂−T,T, we have(91)1τ∑k=1T/τ∫τk−1τk∫ΩGρk−1+ρk−1Vdx−∫ΩGρk+ρkVdxutdt=1τ∫0τ∫ΩGρ0+ρ0Vutdtdx+∫0T−τut+τ−utτ∫ΩGρτ+ρτVdtdx.
Recalling inequalities (88) and (89), we get(92)∫0T∫Ω<ωτ,∇xG′ρτ+V>ρτutdtdx≤1τ∫0τ∫ΩGρ0+ρ0Vutdtdx+∫0T−τut+τ−utτ∫ΩG′ρτ+ρτVdtdx.
We tendτ toward 0, and we obtain(93)limsupτ⟶0∫0,T×Ω<ρτωτ,∇xG′ρτ+V>utdtdx≤∫ΩGρ0+ρ0Vu0dx+∫0,T×ΩG′ρτ+ρVu′tdtdx.
From the strict convexity ofG∈C1]0,∞[, we derive that(94)Gρ0=ρ0G′ρ0−G∗G′ρ0,andGρ=ρG′ρ−G∗Gρ,where G∗ is a Legendre transform of G.
We conclude the proof of claim 2 by using (92) in (93).
Claim 3:(95)∫0,T×Ω<ρω,∇xG′ρ+V>utdtdx≥∫Ωρ0G′ρ0−G∗G′ρ0u0dx+∫0,T×ΩρGρ−G∗G′ρu′tdtdx+∫0,T×ΩρVu′tdtdx+∫Ωρ0Vu0dx.Proof 6.
Defineψt,x=G′ρ+Vut, for t,x∈0,T×Ω.
ψ∈W1,qx0,T×Ω, and then we approximate ψ by Cc∞ℝ×ℝd-functions, and (70) becomes(96)1h∑k=1T/h∫hk−1hk∫Ωρk−1−ρkψt,xdtdx−∫0,T×Ω<ρhωh,∇xG′ρ+V>utdtdx=oh,where oh converges to 0 when h tends to 0. As in (89), we have(97)1h∑k=1T/h∫hk−1hk∫Ωρk−1−ρkψt,xdtdx=1h∫0h∫Ωρ0ψt,xdtdx+∫hT∫Ωρh∂t−hψt,xdtdx,where ∂t−hψt,x=ψt,x−ψt−h,x/h.
Using (96), (97) becomes(98)1h∫0h∫Ωψt,xρ0dtdx+∫hT∫Ωρh∂t−hψt,xdtdx−∫0,T×Ω<ρhωh,∇xG′ρ+V>utdtdx=oh.
We tendh to 0 in (98) and use the fact that ρh converges strongly to ρ in Lqx0,T×Ω and ωh weakly to ω in Lqx0,T×Ω to get(99)∫Ωρ0G′ρ0+Vu0dx+limh⟶0∫0T∫Ωρ∂t−hψt,xdtdx−∫0,T×Ω<ρω,∇xG′ρ+V>utdtdx=0.
By using(100)ρ∂−hψt,x=ρG′ρ∂t−hut+ρV∂t−hut+1hρut−hG′ρ−G′ρt−h,x(101)ρG′ρ−G′ρt−h,x≥G∗G′ρ−G∗G′ρt−h,x,we have(102)ρ∂−hψt,x≥ρG′ρ∂t−hut+ρV∂t−hut+1hut−hG∗G′ρ−G∗G′ρt−h,x=ρG′ρ−G∗G′ρ∂t−hut+ρV∂t−hut+1hG∗G′ρut−G∗G′ρt−h,xut−h.
Next, after integration of the inequality above, we have(103)∫0T∫Ωρ∂−hψt,xdtdx≥∫0T∫ΩρG′ρ−G∗G′ρ∂t−hutdtdx+∫0T∫ΩρV∂t−hutdtdx+1h∫0T∫ΩG∗G′ρut−G∗G′ρt−h,xut−hdtdx.
Note that(104)1h∫0T∫ΩG∗G′ρut−G∗G′ρt−h,xut−hdtdx=1h∫0T∫ΩG∗G′ρutdtdx−1h∫0T∫ΩG∗G′ρt−h,xut−hdtdx,(105)1h∫0T∫ΩG∗G′ρt−h,xut−hdtdx=1h∑k=1T/h∫hk−1hk∫ΩG∗G′ρt−h,xut−hdtdx.
Using the change of variables=t−h in (105), we obtain(106)∫hk−1hk∫0T∫ΩG∗G′ρt−h,xut−hdtdx=∫hk−2hk−1∫0T∫ΩG∗G′ρt−h,xut−hdtdx,for allk≥2.
We combine (106) and (105) to deduce that(107)1h∫0T∫ΩG∗G′ρt−h,xut−hdtdx=1h∫0h∫ΩG∗G′ρt−h,xut−hdtdx+1h∑k=2T/h∫hk−2hk−1∫ΩG∗G′ρt−h,xutdtdx=1h∫0h∫ΩG∗G′ρt−h,xut−hdtdx+1h∫0T−h∫ΩG∗G′ρt−h,xutdtdx.
Now, we use (107) in (108) to derive(108)1h∫0T∫ΩG∗G′ρut−G∗G′ρt−h,xut−hdtdx=−1h∫0h∫ΩG∗G′ρt−h,xut−h+1h∫T−hT∫ΩG∗ρt,xutdtdx.
So, (103) becomes(109)∫0T∫Ωρ∂−hψt,xdtdx≥∫0T∫ΩρG′ρ−G∗G′ρ∂t−hutdtdx+∫0T∫ΩρV∂t−hutdtdx−1h∫0h∫ΩG∗G′ρut−hdtdx+1h∫T−hT∫ΩG∗G′ρt,xutdtdx.
Sincesuppu⊂−T,T, then(110)limh⟶01h∫T−hT∫ΩG∗G′ρt,xutdtdx=∫ΩG∗G′ρT,xuTdx=0.
We deduce that(111)limh⟶0∫0T∫Ωρ∂−hψt,xdtdx≥∫0T∫ΩρG′ρ−G∗G′ρu′tdtdx+∫0T∫ΩρVu′tdtdx−∫ΩG∗G′ρ0u0dx.
Now, we combine (99) and (111) to obtain(112)∫0T∫ΩρG′ρ−G∗G′ρu′tdtdx+∫0T∫ΩρVu′tdtdx−∫ΩG∗G′ρ0u0dx+∫Ωρ0G′ρ0+Vu0dx−∫0,T×Ω<ρω,∇xG′ρ+V>utdtdx≤0.
Finally, we conclude that(113)∫0T∫ΩρG′ρ−G∗G′ρu′tdtdx+∫0T∫ΩρVu′tdtdx+∫Ωρ0Vu0dx+∫Ωρ0G′ρ0−G∗G′ρ0u0dx∫0,T×Ω<ρω,∇xG′ρ+V>utdtdx.
This completes the proof of claim 3.
Now, we are ready to show that the sequencedivxρhωhh converges weakly to divxρ∇xG′ρ+Vqx−2∇xG′ρ+V in Cc∞ℝ×ℝd′. Let ψ∈Cc∞ℝd and u∈Cc∞ℝ be some positive functions such that suppu⊂−T,T, with 0<T<∞. For ϵ≠0, we define ψϵt,x=G′ρ+V−ϵψ, and then(114)∫0,T×Ωρh<ωh−∇xψϵqx−2∇xψϵ,∇xG′ρh+V−∇xψϵ>utdtdx≥0.
So, we obtain(115)∫0,T×Ωρh<ωh,∇xG′ρh+V<utdtdx−∫0,T×Ωρh<ωh,∇xψϵ>utdtdx+∫0,T×Ωρh<∇xψϵqx−2∇xψϵ,∇xψϵ−∇xG′ρh+V>utdtdx≥0.
We use Lemma3 and we tend h to 0 in (115), and then we get(116)∫0,T×Ωρ<ωh,∇xG′ρh+V<utdtdx−∫0,T×Ωρ<ωh,∇xψϵ>utdtdx+∫0,T×Ωρ<∇xψϵqx−2∇xψϵ,∇xψϵ−∇xG′ρh+V>utdtdx≥0.
So,(117)∫0,T×Ωρ<ω−∇xψϵqx−2∇xψϵ,ϵ∇xψ>utdtdx≥0.
Dividing (116) by ϵ>0 and tending ϵ to 0, we reach(118)∫0,T×Ωρ<ω−∇xG′ρVqx−2∇xG′ρV,∇xψ>utdtdx≥0.
By replacingψ by −ψ, we have(119)∫0,T×Ωρ<ω−∇xG′ρVqx−2∇xG′ρV,∇xψ>utdtdx=0,for all test functions ψ∈Cc∞ℝd and u∈Cc∞ℝ such that suppu⊂−T,T.
Thus, we conclude thatdivxρhωhh converges weakly to divxρ∇xG′ρVqx−2∇xG′ρV in Cc∞ℝ×ℝd′.


### 3.3. Proof of the Theorem of the Laplacian Parabolic Equation

Here, we use the strong convergence of the sequenceρhh to ρ in Lqx0,T×Ω and the weak convergence of the nonlinear term divxρhωhh to divxρ∇xG′ρVqx−2∇xG′ρV in Cc∞ℝ×ℝd′, and we establish that ρt,x is a weak solution of the q(x)-Laplacian parabolic equation. Let ψ∈Cc∞ℝ×ℝd be a test function such that suppψ.,x⊂−T,T. By using (71), we obtain(120)1h∑k=1T/h∫hk−1,hk×Ωρk−1−ρkψt,xdtdx−∫0,T×Ωρh<ωh,∇xψ>dtdx=0h,where oh tends to 0 when h tends to 0.Note that(121)1h∑k=1T/h∫hk−1,hk×Ωρk−1−ρkψt,xdtdx=1h∫0h∫Ωρ0ψt,xdtdx+∫hT−h∫Ωρh∂t−hψt,xdtdx.Then, (120) becomes(122)1h∫0h∫Ωρ0ψt,xdtdx+∫hT−h∫Ωρh∂t−hψt,xdtdx−∫0,T×Ωρh<ωh,∇xψ>dtdx=oh.Finally, we tendh to 0 in (122), and we use the fact that ρh converges strongly to ρ and that divxρhωhh converges weakly to divxρ∇xG′ρVqx−2∇xG′ρV:(123)∫Ωρ0ψ0,xdx+∫0,T×Ωρ∂ψ∂tdtdx−∫0,T×Ωρ<∇xG′ρ+Vqx−2∇xG′ρ+V,∇xψ>dtdx=0.We deduce that(124)∫0,T×Ωρ<∂ψ∂t−∇xG′ρ+Vqx−2∇xG′ρ+V,∇xψ>dtdx=−∫Ωρ0ψ0,xdx.It follows thatρt,x is a weak solution of a nonlinear q(x)-Laplacian parabolic equation.


## 3.1. Euler–Lagrangian Equation for Variational Problem (37)

Here, we show that the sequenceρkk defined in (37) is a time discretization of the q(x)-Laplacian parabolic equation, i.e., for all test functions, ϕ∈Cc∞ℝd,ℝ, we have(28)∫Ωρk−ρk−1ϕdx+h∫Ω<∇xG′ρk+Vqx−2∇xG′ρk+V,∇xϕ>ρkdx=Akhϕ,where Akhϕ converges to 0 when h tends to 0.Lemma 1.
Assume thatHρ0, HG, and HV1 hold. Then, the variational problem(29)P1:infρ∈PΩIρ≔∫ΩGρ+ρVdx+Kp.hρ0,ρ,where(30)Kp.hρ0,ρ≔infγ∈Πρ0,ρ∫Ω×Ωx−ypxpxhpx−1dγ,admits a unique solution ρ1 in PΩ which satisfies N≤ρ1≤M.Proof 1.
SinceN≤ρ0≤M, then the solution ρ1 of variational problem P1 (if there exists) satisfies N≤ρ1≤M. The proof of the maximum principle is carried out similarly as given in [7].
Since∫ΩGρ0dx<∞ and V∈L∞Ω, we have Iρ0<∞.
Letρ∈PΩ; since G is convex and V∈L∞Ω,(31)Iρ≥∫ΩGρ+ρVdx≥ΩG1Ω−VL∞Ω.
We conclude thatinfρ∈PΩIρ is finite.
Letρnn be a sequence in PΩ such that N≤ρn≤M and Iρn converges to infρ∈PΩIρ. Then, ρnn converges weakly∗ to ρ1 in L∞Ω up to a subsequence.
G being positive and convex, then(32)∫ΩGρn+ρnVdx≥∫ΩGρ1+ρ1Vdx+∫ΩG′ρ1ρn−ρ1dx+∫Ωρn−ρ1Vdx.
Next, we use the fact thatG′ρ1∈L∞Ω and V∈L∞Ω to obtain(33)liminfn⟶∞∫ΩGρn+ρnVdx≥∫ΩGρ1+ρ1Vdx.
Since the variable exponentp. is continuous on Ω, the Kantorovich problem(34)K:infγ∈Πρ0,ρn∫Ω×Ωx−ypxpxhpx−1dγ,admits a solution γn. Moreover, since Ω is bounded, the sequence γnn converges to a measure γ1 in PΩ×Ω narrowly, and γ1∈Πρ0,ρ1; and then, we derive that(35)liminfn⟶∞Kp.hρ0,ρn≥∫Ω×Ωx−ypxpxhpx−1dγ1≥Kp.hρ0,ρ1.
We combine (35) and (33) to obtain(36)liminfn⟶∞Iρn≥Iρ1.
Thus,ρ1 is a solution of variational problem P1. From the strict convexity of G, we deduce that ρ⟼∫ΩGρ+ρVdx is strictly convex and so is ρ⟼Iρ on PΩ, and consequently, the uniqueness of the solution ρ1 of P1 follows.Now, we assume thatHρ0, HG, and HV1 hold. Then, from Lemma 1, we obtain that the variational problem(37)Pk:infρ∈PΩIρ≔∫ΩGρ+ρVdx+Kp.hρk−1,ρ,with(38)Kp.hρk−1,ρ≔infγ∈Πρk−1,ρ∫Ω×Ωx−ypxpxhpx−1dγ,admits a unique solution ρk for all k≥1.Next, we prove thatρkk is a time discretization of the nonlinear q(x)-Laplacian parabolic equation. In order to achieve this, we use the following lemma.Lemma 2.
Assume thatHρ0, HG, and HV1 hold. Then, the Kantorovich problem(39)K:infγ∈Πρk−1,ρk∫Ω×Ωx−ypxpxhpx−1dγ,admits a solution γk such that(40)suppγk⊂x,y∈Ω×Ω,y=x−h∇xG′ρk+Vqx−2∇xG′ρk+V,where qx=px/px−1.Proof 2.
The proof of Lemma2 is derived following the two steps.Step 1.
We first assume thatV∈C2Ω.
Fixψ∈Cc∞ℝd,ℝd. Let Tϵϵ∈ℝ in C∞ℝd,ℝd defined by(41)Tϵx=x+ϵψx.
Define the probability densityρϵ as ρϵ=Tϵ#ρk and γϵ as a probability measure on Ω×Ω defined by(42)∫Ω×Ωϕx,ydγϵ=∫Ω×Ωϕx,Tϵydγk,for all ϕ∈Cb0Ω×Ω.
Forϵ≠0,(43)Iρϵ−Iρkϵ=∫ΩGρϵ−Gρkϵdx+∫ΩVTϵ−Vxϵdx+1ϵKp.hρϵ,ρk−1−Kp.hρk−1,ρk.
SinceG satisfies HG, then(44)limϵ⟶0∫ΩGρϵ−Gρkϵdx=∫Ω∇xG′ρk.ψρkdx.
SinceV∈C2Ω, from the Taylor formula, we have (see [7])(45)VTϵx−Vx=ϵ∇xVx⋅ψx+ϵ2ψxτ∇x2Vx+ϵθψx⋅ψx,withθ∈0,1.
Then, we have after integration,(46)∫ΩVTϵ−Vxϵdx=∫Ω<∇Vx,ψ>ρkdx+ϵ∫Ωψxτ⋅∇2Vx+ϵθψx⋅ψxρkdx.
By using (41) and the fact that V∈W1,∞Ω, we have(47)ψxτ.∇2Vx+ϵθψx.ψx≤∇2VL∞ΩψL∞Ω.
We now use the dominated convergence theorem to have(48)limϵ⟶0∫ΩVTϵ−Vxϵdx=∫Ω∇V.ψρkdx.
Note thatγϵ defined in (42) belongs to Πρk−1,ρϵ, and(49)Kp.hρk−1,ρk=∫Ω×Ωx−ypxpxhpx−1dγk,Kp.hρk−1,ρϵ≤∫Ω×Ωx−Tϵypxpxhpx−1dγk.
So, forϵ>0, we have(50)1ϵKp.hρk−1,ρϵ−Kp.hρk−1,ρk≤1ϵ∫Ω×Ωx−Tϵypx−x−ypxpxhpx−1dγk.
Note that(51)limϵ⟶0+1ϵ∫Ω×Ωx−Tϵy|px−|x−y|pxpxhpx−1dγk=−∫Ω×Ω<|x−yh|px−2x−yh,ψ>dγk.
Indeed,(i)
We have(52)limϵ⟶0+1ϵx−Tϵypx−x−ypxpxhpx−1=−x−yhpx−2x−yh.ψ.(ii)
On the contrary, the Taylor formula with respect toϵ enables us to write(53)x−Tϵypx−x−ypxpxhpx−1=−ϵx−y−θϵψyhpx−2x−y−θϵψyh.ψy,where θ∈0,1 and Tϵy=y+ϵψy. Also, we have(54)x−Tϵypx−x−ypxϵpxhpx−1=x−y−θϵψyhpx−1ψy≤x−y−θϵψyh+1px−1ψy≤x−y−θϵψyh+1pxψy≤2p+−1x−yhpx+2p+−1ψyh+1p+ψy,with(55)∫Ω×Ω2p+−1x−yhpx+2p+−1ψyh+1p+ψydγkx,y<∞.
We then combine the results given in (i) and (ii) and the dominated convergence theorem to obtain (50).
Sinceρk minimizes I on PΩ, then(56)limϵ⟶0Iρϵ−Iρkϵ=0.
So, we combine (44), (48), and (50) to obtain that(57)0=limϵ⟶0+Iρϵ−Iρkϵ≤∫Ω∇xG′ρk+V.ψρkdx−∫Ω×Ω<x−yhpx−2x−yh,ψ>dγk.
Replacingψ by −ψ in (56), we get(58)∫Ω<∇xG′ρk+V,ψ>ρkdx−∫Ω×Ω<x−yhpx−2x−yh,ψ>dγk≤0.
From (56) and (57), we deduce that(59)∫Ω<∇xG′ρk+V,ψ>ρkdx+∫Ω×Ω<y−xhpx−2y−xh,ψ>dγk=0.
Thus, we obtain (40) when V∈C2Ω.
Now, let us establish the proof of the lemma whenV∈W1,∞Ω and is nonregular.Step 2.
Assume thatV∈W1,∞Ω. Let Vδδ be a sequence in C2Ω such that limδ⟶0Vδ−VW1,∞Ω=0.
By fixingδ>0, we define the sequence ρkδk such that ρ0δ=ρ0 with ρkδ (for k≥1), the solution of the variational problem(60)Pkδ:infρ∈PΩ∫ΩGρ+ρVδdx+Kp.hρk−1δ,ρ,where Kp.hρk−1δ,ρ is defined as in (38). As in Lemma 1, the variational problem Pkδ admits a unique solution ρkδ in PΩ, and N≤ρkδ≤M. Hence, the Kantorovich problem(61)Kδ:infγ∈Πρk−1δ,ρkδ∫Ω×Ωx−ypxpxhpx−1dγ,admits a solution γkδ such that(62)suppγkδ⊂x,y∈Ω×Ω,y=x−h∇xG′ρkδ+Vδqx−2∇xG′ρkδ+Vδ.
Let us show thatρkδδ converges to ρk and γkδδ converges to γk up to a subsequence, as well as suppγk satisfies (40)
UsingPkδ and (61), we have(63)∫ΩGρk−1δ+ρk−1δVδdx−∫ΩGρkδ+ρkδVδdx≥h∫Ω∇xG′ρkδ+Vδqxpxρkδdx.
However,G is convex, and V∈W1,∞Ω; and then,(64)∫Ω∇xG′ρkδ+Vδqxρkδ≤p+h∫ΩGρ0dx−ΩG1Ω+2VL∞Ω.
Furthermore, recalling (63) and the fact that N≤ρkδ≤M, ∇xG′ρkδ=G′′ρkδ∇xρkδ, and ∇xG′ρkδqx≤2q+−1∇xG′ρkδ+Vδqx+∇Vδqx, we obtain(65)∫Ω∇xρkδqxdx≤2q+−1k0p+Nh∫ΩGρ0dx−ΩG1Ω+2VL∞Ω+∇VL∞Ωq+,where k0=inft,x∈N,M×ΩG′′tqx. We conclude that ρkδδ is bounded on W1,qxΩ. Since q. is continuous and Ω is a bounded set, the injection W1,qxΩ↪↪LqxΩ is compact (see [9]), and hence, the sequence ρkδδ converges strongly to some uk in LqxΩ up to a subsequence. Moreover, ρkδ minimizes Pkδ for k∈ℕ∗, and then, we have(66)∫ΩGρ+ρVδdx+Kp.hρk−1δ,ρ≥∫ΩGρkδ+ρkδVδdx+Kp.hρk−1δ,ρkδ,for all ρ∈PΩ. Also, from the boundedness of Ω, γkδδ converges narrowly to a measure Γk in PΩ×Ω, and Γk∈Πuk−1,uk. Then, we use the strong convergence of ρkδδ to uk and the fact that Vδ−VW1,∞Ω converges to 0 when δ⟶0 to obtain that(67)∫ΩGρ+ρVdx+Kp.huk−1,ρ≥∫ΩGuk+ukVdx+Kp.huk−1,uk,for all ρ∈PΩ. Since ρ0δ=ρ0, we conclude that uk=ρk and Γk=γk for all k∈ℕ∗.
By using (63) and N≤ρk≤M, we deduce that ∇xG′ρkδ+Vδδ is bounded in LqxΩd. Since ρkδδ converges strongly to ρk in LqxΩ and G′ is continuous, then ∇xG′ρkδ+Vδδ converges weakly to ∇xG′ρk+V in LqxΩd.
Next, we use (57), and we have(68)∫Ω<∇xG′ρkδ+Vδ,ψ>ρkδdx+∫Ω×Ω<y−xhpx−2y−xh,ψ> dγkδ=0.
Sinceρkδδ converges strongly to ρk in LqxΩ, ∇xG′ρkδ+Vδδ converges weakly to ∇xG′ρk+V in LqxΩd, and γkδδ converges narrowly to γk in PΩ×Ω; then, tending δ to 0 in (67), we obtain(69)∫Ω<∇xG′ρk+Vδ,ψ>ρkdx+∫Ω×Ω<y−xhpx−2y−xh,ψ> dγk=0,for all test functions, ψ∈Cc∞ℝd,ℝd.
Finally, we obtain the equality(70)y=x−h∇xG′ρk+Vqx−2∇xG′ρk+Vγka.e.
Now, let us prove thatρkk is a time discretization of a nonlinear q(x)-Laplacian parabolic equation. Let ψ∈Cc∞ℝd,ℝ be a test function; we have(71)∫Ωρk−ρk−1ψdx=∫Ω×Ωψy−ψxdγkx,y.
We now use (69) and the Taylor formula to obtain(72)∫Ωρk−ρk−1ψdx=−h∫Ω<∇xG′ρk+Vqx−2∇xG′ρk+V,∇ψ>ρkdx+h2∫Ω×Ω<∇xG′ρk+Vτ,∇2ψx+θy−x∇xG′ρk+V>dγkx,y,where θ∈0,1 and ∇xG′ρk+Vτ is the transpose of ∇xG′ρk+Vτ.
Let us show that(73)Akhψ=h2∫Ω×Ω<∇xG′ρk+Vτ,∇2ψx+θy−x∇xG′ρk+V>dγkx,yconverge to 0 when h tends to 0. Since qx−1=qx/px, we have(74)Akhψ≤h2∇2ψL∞Ω∫Ω∇xG′ρk+V2qx/pxρkdx.
DefineΩ1≔x∈Ω,px≥2 and Ω1≔x∈Ω,px<2:(75)∫Ω1∇xG′ρk+V2qx/pxρkdx≤2q+−1∫Ω∇xG′ρk+Vqxρkdx+1,∫Ω2∇xG′ρk+V2qx/pxρkdx≤h1−2p−∫Ω∇xG′ρk+Vqxρkdx.
We use (64) and (73) in (72), and we have(76)Akhψ≤2p+−1k0h2∇xψL∞Ω+p+h∇xψL∞Ω+h2−2/p−∫ΩGρ0dx−ΩG1Ω+2∇L∞Ω.
We tendh to 0, and we conclude that Akhψ converges to 0. Thus, the sequence ρkk is a time discretization of the nonlinear q(x)-Laplacian parabolic equation.
Defineρht,x:0,+∞×Ω⟶0,+∞ as follows:(77)ρht,x=ρkxift∈k−1h,kh,andρh0,x=ρ0x.
We prove in the next section that the sequenceρhh converges strongly to a function ρt,x in Lqx0,T×Ω for 0<T<∞ and that the nonlinear term sequence divxρh∇xG′ρh+Vqx−2∇xG′ρh+Vh converges weakly to divxρh∇xG′ρh+Vqx−2∇xG′ρh+Vh in ψ∈Cc∞ℝd,ℝd.


## 3.2. Strong Convergence ofρhh and Weak Convergence of the Nonlinear Term Sequence

Here, we denote the nonlinear term sequence bydivxρhωhh, with ωh≔|∇xG′ρh+Vqx−2∇xG′ρh+V.SinceN≤ρk≤M, then N≤ρh≤M. Thus, we use (64), and we have(78)∫0,T×Ω∇xρhqxdx≤2q+−1k0p+N∫ΩGρ0dx−ΩG1Ω+2VL∞Ω+∇VL∞Ωq+,where 0<T<∞. We conclude that ρht,.h is bounded in W1,qxΩ for almost all t∈0,T. By using the compact injection W1,qxΩ↪↪LqxΩ, we deduce that ρht,.h converges strongly to a function ρt,. in LqxΩ. Since N≤ρh≤M and 0,T×Ω is bounded, we use the dominated convergence theorem to have(79)limh⟶0∫0T∫Ωρht,x−ρt,xqxdxdt=0.Then, up to a subsequence, the sequenceρhh converges strongly to ρt,x in Lqx0,T×Ω. By using (64) and N≤ρh≤M, we deduce that ∇xG′ρh+Vh is bounded in Lqx0,T×Ωd. Then, since G′ is continuous and ρhh converges strongly to ρ in Lqx0,T×Ω, we conclude that ∇xG′ρh+Vh converges weakly to ∇xG′ρ+V in Lqx0,T×Ωd.Note thatωhpx=∇xG′ρh+Vqx. Then, using the fact that ∇xG′ρh+Vh is bounded in Lqx0,T×Ω, we conclude that ωhh is bounded in Lqx0,T×Ω. Thus, the sequence ωhh converges weakly to some ω∈Lpx0,T×Ω.As in [7], we derive easily that divxρh∇xG′ρh+Vqx−2∇xG′ρh+Vh converges weakly to divxρ∇xG′ρ+Vqx−2∇xG′ρ+V in ψ∈Cc∞ℝd,ℝd.Lemma 3.
FixT>0. Let u∈Cc∞ℝ be a test function such that suppu⊂−T,T. We have(80)limh⟶0∫0,T×Ω<ρhωh,∇xG′ρh+V>utdtdx=∫0,T×Ω<ρω,∇xG′ρh+V>utdtdx.Proof 3.
The proof of this lemma is achieved via the following three claims:
Claim 1:(81)liminfh⟶0∫0,T×Ωρh<ωh,∇xG′ρh+V>utdtdx≥∫0,T×Ωρ<ω,∇xG′ρh+V>utdtdx.Proof 4.
Sinceρh≥0, u≥0, and v⟼vqx−2v is monotone for all x fixed in Ω, we have(82)∫0,T×Ωρh<ωh−∇xG′ρ+Vqx−2∇xG′ρ+V,∇xG′ρh+V−∇xG′ρ+V>utdtdx≥0.
Using (81), we have(83)∫0,T×Ω<ρhωh,∇xG′ρh+V>utdtdx≥∫0,T×Ω<ρhωh,∇xG′ρ+V>utdtdx+∫0,T×Ω<ρh∇xG′ρ+Vqx−2∇xG′ρ+V,∇xG′ρh+V−∇xG′ρ+V>utdtdx.
Sinceρhh converges strongly to ρ in Lqx0,T×Ω, ∇xG′ρh+Vh converges weakly to ∇xG′ρ+V in Lqx0,T×Ω, and ωhh converges weakly to ω in Lqx0,T×Ω, then(84)liminfh⟶0∫0,T×Ω<ρhωh,∇xG′ρ+V>utdtdx=∫0,T×Ω<ρω,∇xG′ρ+V>utdtdx,(85)liminfh⟶0∫0,T×Ω<ρh∇xG′ρ+Vqx−2∇xG′ρ+V,∇xG′ρh+V−∇xG′ρ+V>utdtdx=0.
We now use (83) and (84) in (82) to conclude the proof of claim 1.
Claim 2:(86)limsuph⟶0∫0,T×Ω<ρhωh,∇xG′ρh+V>utdtdx≤∫Ωρ0G′ρ0−G∗G′ρ0u0dx+∫0,T×ΩρG′ρ−G∗G′ρhu′tdtdx+∫0,T×ΩρVu′tdtdx+∫Ωρ0Vu0dx.Proof 5.
We use the descent algorithmPk and (70) to obtain(87)∫ΩGρk−1+ρk−1Vdx−∫ΩGρk+ρkVdx≥h∫Ω∇xG′ρk+Vqxpxρkdx.
Note that<ωk,∇xG′ρk+V>=∇xG′ρk+Vqx, with ωk=∇xG′ρk+Vqx−2∇xG′ρk+V. Then, we obtain after integration over 0,T that(88)1h∑k=1T/h∫hk−1hk∫ΩGρk−1+ρk−1Vdx−∫ΩGρk+ρkVdxutdt≥∫0T∫Ω<ωk,∇xG′ρk+V>pxρkutdtdx.
We usepx≤p+ and τ=h/p+ in (87), and we have(89)1h∑k=1T/h∫hk−1hk∫ΩGρk−1+ρk−1Vdx−∫ΩGρk+ρkVdxutdt≥∫0T∫Ω<ωτ,∇xG′ρτ+V>ρτutdtdx,where ρτ is defined as follows:(90)ρτt,x=ρkx,ift∈τk−1,τk,ρτ0,x=ρ0x,and ωτ=∇xG′ρτ+Vqx−2∇xG′ρτ+V.
Using the fact thatsuppu⊂−T,T, we have(91)1τ∑k=1T/τ∫τk−1τk∫ΩGρk−1+ρk−1Vdx−∫ΩGρk+ρkVdxutdt=1τ∫0τ∫ΩGρ0+ρ0Vutdtdx+∫0T−τut+τ−utτ∫ΩGρτ+ρτVdtdx.
Recalling inequalities (88) and (89), we get(92)∫0T∫Ω<ωτ,∇xG′ρτ+V>ρτutdtdx≤1τ∫0τ∫ΩGρ0+ρ0Vutdtdx+∫0T−τut+τ−utτ∫ΩG′ρτ+ρτVdtdx.
We tendτ toward 0, and we obtain(93)limsupτ⟶0∫0,T×Ω<ρτωτ,∇xG′ρτ+V>utdtdx≤∫ΩGρ0+ρ0Vu0dx+∫0,T×ΩG′ρτ+ρVu′tdtdx.
From the strict convexity ofG∈C1]0,∞[, we derive that(94)Gρ0=ρ0G′ρ0−G∗G′ρ0,andGρ=ρG′ρ−G∗Gρ,where G∗ is a Legendre transform of G.
We conclude the proof of claim 2 by using (92) in (93).
Claim 3:(95)∫0,T×Ω<ρω,∇xG′ρ+V>utdtdx≥∫Ωρ0G′ρ0−G∗G′ρ0u0dx+∫0,T×ΩρGρ−G∗G′ρu′tdtdx+∫0,T×ΩρVu′tdtdx+∫Ωρ0Vu0dx.Proof 6.
Defineψt,x=G′ρ+Vut, for t,x∈0,T×Ω.
ψ∈W1,qx0,T×Ω, and then we approximate ψ by Cc∞ℝ×ℝd-functions, and (70) becomes(96)1h∑k=1T/h∫hk−1hk∫Ωρk−1−ρkψt,xdtdx−∫0,T×Ω<ρhωh,∇xG′ρ+V>utdtdx=oh,where oh converges to 0 when h tends to 0. As in (89), we have(97)1h∑k=1T/h∫hk−1hk∫Ωρk−1−ρkψt,xdtdx=1h∫0h∫Ωρ0ψt,xdtdx+∫hT∫Ωρh∂t−hψt,xdtdx,where ∂t−hψt,x=ψt,x−ψt−h,x/h.
Using (96), (97) becomes(98)1h∫0h∫Ωψt,xρ0dtdx+∫hT∫Ωρh∂t−hψt,xdtdx−∫0,T×Ω<ρhωh,∇xG′ρ+V>utdtdx=oh.
We tendh to 0 in (98) and use the fact that ρh converges strongly to ρ in Lqx0,T×Ω and ωh weakly to ω in Lqx0,T×Ω to get(99)∫Ωρ0G′ρ0+Vu0dx+limh⟶0∫0T∫Ωρ∂t−hψt,xdtdx−∫0,T×Ω<ρω,∇xG′ρ+V>utdtdx=0.
By using(100)ρ∂−hψt,x=ρG′ρ∂t−hut+ρV∂t−hut+1hρut−hG′ρ−G′ρt−h,x(101)ρG′ρ−G′ρt−h,x≥G∗G′ρ−G∗G′ρt−h,x,we have(102)ρ∂−hψt,x≥ρG′ρ∂t−hut+ρV∂t−hut+1hut−hG∗G′ρ−G∗G′ρt−h,x=ρG′ρ−G∗G′ρ∂t−hut+ρV∂t−hut+1hG∗G′ρut−G∗G′ρt−h,xut−h.
Next, after integration of the inequality above, we have(103)∫0T∫Ωρ∂−hψt,xdtdx≥∫0T∫ΩρG′ρ−G∗G′ρ∂t−hutdtdx+∫0T∫ΩρV∂t−hutdtdx+1h∫0T∫ΩG∗G′ρut−G∗G′ρt−h,xut−hdtdx.
Note that(104)1h∫0T∫ΩG∗G′ρut−G∗G′ρt−h,xut−hdtdx=1h∫0T∫ΩG∗G′ρutdtdx−1h∫0T∫ΩG∗G′ρt−h,xut−hdtdx,(105)1h∫0T∫ΩG∗G′ρt−h,xut−hdtdx=1h∑k=1T/h∫hk−1hk∫ΩG∗G′ρt−h,xut−hdtdx.
Using the change of variables=t−h in (105), we obtain(106)∫hk−1hk∫0T∫ΩG∗G′ρt−h,xut−hdtdx=∫hk−2hk−1∫0T∫ΩG∗G′ρt−h,xut−hdtdx,for allk≥2.
We combine (106) and (105) to deduce that(107)1h∫0T∫ΩG∗G′ρt−h,xut−hdtdx=1h∫0h∫ΩG∗G′ρt−h,xut−hdtdx+1h∑k=2T/h∫hk−2hk−1∫ΩG∗G′ρt−h,xutdtdx=1h∫0h∫ΩG∗G′ρt−h,xut−hdtdx+1h∫0T−h∫ΩG∗G′ρt−h,xutdtdx.
Now, we use (107) in (108) to derive(108)1h∫0T∫ΩG∗G′ρut−G∗G′ρt−h,xut−hdtdx=−1h∫0h∫ΩG∗G′ρt−h,xut−h+1h∫T−hT∫ΩG∗ρt,xutdtdx.
So, (103) becomes(109)∫0T∫Ωρ∂−hψt,xdtdx≥∫0T∫ΩρG′ρ−G∗G′ρ∂t−hutdtdx+∫0T∫ΩρV∂t−hutdtdx−1h∫0h∫ΩG∗G′ρut−hdtdx+1h∫T−hT∫ΩG∗G′ρt,xutdtdx.
Sincesuppu⊂−T,T, then(110)limh⟶01h∫T−hT∫ΩG∗G′ρt,xutdtdx=∫ΩG∗G′ρT,xuTdx=0.
We deduce that(111)limh⟶0∫0T∫Ωρ∂−hψt,xdtdx≥∫0T∫ΩρG′ρ−G∗G′ρu′tdtdx+∫0T∫ΩρVu′tdtdx−∫ΩG∗G′ρ0u0dx.
Now, we combine (99) and (111) to obtain(112)∫0T∫ΩρG′ρ−G∗G′ρu′tdtdx+∫0T∫ΩρVu′tdtdx−∫ΩG∗G′ρ0u0dx+∫Ωρ0G′ρ0+Vu0dx−∫0,T×Ω<ρω,∇xG′ρ+V>utdtdx≤0.
Finally, we conclude that(113)∫0T∫ΩρG′ρ−G∗G′ρu′tdtdx+∫0T∫ΩρVu′tdtdx+∫Ωρ0Vu0dx+∫Ωρ0G′ρ0−G∗G′ρ0u0dx∫0,T×Ω<ρω,∇xG′ρ+V>utdtdx.
This completes the proof of claim 3.
Now, we are ready to show that the sequencedivxρhωhh converges weakly to divxρ∇xG′ρ+Vqx−2∇xG′ρ+V in Cc∞ℝ×ℝd′. Let ψ∈Cc∞ℝd and u∈Cc∞ℝ be some positive functions such that suppu⊂−T,T, with 0<T<∞. For ϵ≠0, we define ψϵt,x=G′ρ+V−ϵψ, and then(114)∫0,T×Ωρh<ωh−∇xψϵqx−2∇xψϵ,∇xG′ρh+V−∇xψϵ>utdtdx≥0.
So, we obtain(115)∫0,T×Ωρh<ωh,∇xG′ρh+V<utdtdx−∫0,T×Ωρh<ωh,∇xψϵ>utdtdx+∫0,T×Ωρh<∇xψϵqx−2∇xψϵ,∇xψϵ−∇xG′ρh+V>utdtdx≥0.
We use Lemma3 and we tend h to 0 in (115), and then we get(116)∫0,T×Ωρ<ωh,∇xG′ρh+V<utdtdx−∫0,T×Ωρ<ωh,∇xψϵ>utdtdx+∫0,T×Ωρ<∇xψϵqx−2∇xψϵ,∇xψϵ−∇xG′ρh+V>utdtdx≥0.
So,(117)∫0,T×Ωρ<ω−∇xψϵqx−2∇xψϵ,ϵ∇xψ>utdtdx≥0.
Dividing (116) by ϵ>0 and tending ϵ to 0, we reach(118)∫0,T×Ωρ<ω−∇xG′ρVqx−2∇xG′ρV,∇xψ>utdtdx≥0.
By replacingψ by −ψ, we have(119)∫0,T×Ωρ<ω−∇xG′ρVqx−2∇xG′ρV,∇xψ>utdtdx=0,for all test functions ψ∈Cc∞ℝd and u∈Cc∞ℝ such that suppu⊂−T,T.
Thus, we conclude thatdivxρhωhh converges weakly to divxρ∇xG′ρVqx−2∇xG′ρV in Cc∞ℝ×ℝd′.


## 3.3. Proof of the Theorem of the Laplacian Parabolic Equation

Here, we use the strong convergence of the sequenceρhh to ρ in Lqx0,T×Ω and the weak convergence of the nonlinear term divxρhωhh to divxρ∇xG′ρVqx−2∇xG′ρV in Cc∞ℝ×ℝd′, and we establish that ρt,x is a weak solution of the q(x)-Laplacian parabolic equation. Let ψ∈Cc∞ℝ×ℝd be a test function such that suppψ.,x⊂−T,T. By using (71), we obtain(120)1h∑k=1T/h∫hk−1,hk×Ωρk−1−ρkψt,xdtdx−∫0,T×Ωρh<ωh,∇xψ>dtdx=0h,where oh tends to 0 when h tends to 0.Note that(121)1h∑k=1T/h∫hk−1,hk×Ωρk−1−ρkψt,xdtdx=1h∫0h∫Ωρ0ψt,xdtdx+∫hT−h∫Ωρh∂t−hψt,xdtdx.Then, (120) becomes(122)1h∫0h∫Ωρ0ψt,xdtdx+∫hT−h∫Ωρh∂t−hψt,xdtdx−∫0,T×Ωρh<ωh,∇xψ>dtdx=oh.Finally, we tendh to 0 in (122), and we use the fact that ρh converges strongly to ρ and that divxρhωhh converges weakly to divxρ∇xG′ρVqx−2∇xG′ρV:(123)∫Ωρ0ψ0,xdx+∫0,T×Ωρ∂ψ∂tdtdx−∫0,T×Ωρ<∇xG′ρ+Vqx−2∇xG′ρ+V,∇xψ>dtdx=0.We deduce that(124)∫0,T×Ωρ<∂ψ∂t−∇xG′ρ+Vqx−2∇xG′ρ+V,∇xψ>dtdx=−∫Ωρ0ψ0,xdx.It follows thatρt,x is a weak solution of a nonlinear q(x)-Laplacian parabolic equation.


## 4. Asymptotic Behavior

In this section, we give the proof of Theorem2 which is derived from the following three lemmas.Lemma 4.
Letρ1andρ2 be two probability densities on Ω. Assume that V satisfies HV2. If T#+ρ1=ρ2 is the solution of the Monge problem(125)M+:infT#+ρ1=ρ2∫Ωx−T+xp+ρ1dx,then(126)Eρ1−Eρ2+λ2p+Wp.p+ρ1,ρ2≤∫Ω<x−T+x,∇xG′ρ1+V>ρ1dx.In particular, ifρ∞∈PΩ satisfies ρ∞∇xG′ρ∞+V=0, then(127)Wp.p+ρ,ρ∞≤2p+λEρ−Eρ∞,for all ρ∈PΩ.Here,Wp. is the p.-Wasserstein distance defined in (5).Proof 7.
Letρ1,ρ2∈PΩ. Fix p+=supΩpx>1. It is known from the work in [8] that the Monge problem(128)M+:infT#+ρ1=ρ2∫Ωx−T+xp+ρ1dx,admits a unique solution T+. Therefore, since G and V satisfy, respectively, HG and HV2, then(129)Eρ1−Eρ2+λWp+p+ρ1,ρ2≤∫Ω<x−T+,∇xG′ρ1+V>ρ1dx,where ∫Ωx−T+p+ρ1dx=Wp+p+ρ1,ρ2 and Wp+ is the p+-Wasserstein distance.
The continuous injectionLρ1p+Ω↪Lρ1p.Ω implies that(130)Wp.ρ1,ρ2≤2Wp+ρ1,ρ2.
So, we use (129) and (128) to derive(131)Eρ1−Eρ2+λ2p+Wp.p+ρ1,ρ2≤∫Ω<x−T+,∇xG′ρ1+V>ρ1dx.
In particular, ifρ1=ρ∞∈PΩ satisfies ρ∞∇xG′ρ∞+V=0 and ρ2=ρ∈PΩ, then(132)Eρ∞−Eρ+λ2p+Wp.p+ρ∞,ρ≤0.
So, we have(133)Wp.p+ρ∞,ρ≤2p+λEρ−Eρ∞,for all ρ∈PΩ.
Note thatρ∞ is a stationary solution of the q(x)-Laplacian parabolic equation, which minimizes the functional Eρ=∫ΩGρ+ρVdx on PΩ.Lemma 5.
Letρ1,ρ2∈PΩ. Then,(134)Eρ1−Eρ2≤2p+′λp+′∇xG′ρ1+VLρ1q.Ωp+′,with p+′=p+/p+−1.Proof 8.
We applied Young’s inequality to obtain(135)<x−T+,∇xG′ρ1+V>≤μx−T+p++1μp+′∇xG′ρ1+Vp+′,where μ>0 and p+′=p+/p+−1. Then,(136)∫Ω<x−T+,∇xG′ρ1+V>ρ1dx≤μ∫Ωx−T+p+ρ1dx+1μp+′∫Ω∇xG′ρ1+Vp+′ρ1dx,and since p+′≤qx, we use the continuous injection Lρ1q.Ω↪Lρ1p+′Ω, and we get(137)∫Ω∇xG′ρ1+Vp+′ρ1dx≤2p+′∇xG′ρ1+VLρ1q.Ωp+′.
Now, we combine (128), (137), and (136) to obtain(138)Eρ1−Eρ2+λ−μWp+p+ρ1,ρ2≤2p+′μp+′∇xG′ρ1+VLρ1q.Ωp+′.
In particular, ifμ=λ, we obtain (135).Lemma 6.
(139)∇xG′ρ1+VLρ1q.Ωp+′≤Max∫Ω∇xG′ρ1+Vqxρ1dx,∫Ω∇xG′ρ1+Vqxρ1dx1/q+.Proof 9.
From [9], we get(140)∇xG′ρ1+VLρ1q.Ωq−≤∫Ω∇xG′ρ1+Vqxρ1dx,if∇xG′ρ1+VLρ1q.Ω≥1,and(141)∇xG′ρ1+VLρ1q.Ωp+′≤∫Ω∇xG′ρ1+Vqxρ1dx,if∇xG′ρ1+VLρ1q.Ω<1.
Since1<p+′≤q−, we have(142)∇xG′ρ1+VLρ1q.Ωp+′≤∇xG′ρ1+Vqxρ1dx,if∇xG′ρ1+VLρ1q.Ω≥1,(143)∇xG′ρ1+VLρ1q.Ωp+′≤∇xG′ρ1+VLρ1q.Ω,if∇xG′ρ1+VLρ1q.Ω<1.
By using (141), we derive that(144)∇xG′ρ1+VLρ1q.Ω≤∫Ω∇xG′ρ1+Vqxρ1dx1/q+,if∇xG′ρ1+VLρ1q.Ω<1.
Next, we combine (143) and (144) to derive(145)∇xG′ρ1+VLρ1q.Ωp+′≤∫Ω∇xG′ρ1+Vqxρ1dx1/q+,if∇xG′ρ1+VLρ1q.Ω<1.
By using (142) and (143), we obtain (140).


### 4.1. Proof of Theorem2

Letρt,x be a solution of the q(x)-Laplacian parabolic equation. We have(146)ddtEρt,.−Eρ∞=−∫Ω∇xG′ρt,x+Vqxρt,xdx.Define(147)I1≔t∈0,+∞,∇xG′ρt,.+VLρt,.q.Ω≥1,(148)I2≔t∈0,+∞,∇xG′ρt,.+VLρt,.q.Ω<1.We combine (135), (142), and (146) to get(149)ddtEρt,.−Eρ∞≤−λ2p+′Eρt,.−Eρ∞,on I1, and we combine (135), (145), and (146) to have(150)ddtEρt,.−Eρ∞≤−λ2p+′q+Eρt,.−Eρ∞q+,on I2.Next, we have after integration,(151)Eρt,.−Eρ∞≤Eρt2,.−Eρ∞1+λ/2p+′q+q+−1Eρt2,.−Eρ∞q+−1t−t21/q+−1,on I2, and(152)Eρt,.−Eρ∞≤e−λ/2p+′t−t1Eρ0−Eρ∞,on I1.To conclude, we use Lemma4 and expressions (151) and (152), and then, the proof of Theorem 2 is complete.


## 4.1. Proof of Theorem2

Letρt,x be a solution of the q(x)-Laplacian parabolic equation. We have(146)ddtEρt,.−Eρ∞=−∫Ω∇xG′ρt,x+Vqxρt,xdx.Define(147)I1≔t∈0,+∞,∇xG′ρt,.+VLρt,.q.Ω≥1,(148)I2≔t∈0,+∞,∇xG′ρt,.+VLρt,.q.Ω<1.We combine (135), (142), and (146) to get(149)ddtEρt,.−Eρ∞≤−λ2p+′Eρt,.−Eρ∞,on I1, and we combine (135), (145), and (146) to have(150)ddtEρt,.−Eρ∞≤−λ2p+′q+Eρt,.−Eρ∞q+,on I2.Next, we have after integration,(151)Eρt,.−Eρ∞≤Eρt2,.−Eρ∞1+λ/2p+′q+q+−1Eρt2,.−Eρ∞q+−1t−t21/q+−1,on I2, and(152)Eρt,.−Eρ∞≤e−λ/2p+′t−t1Eρ0−Eρ∞,on I1.To conclude, we use Lemma4 and expressions (151) and (152), and then, the proof of Theorem 2 is complete.


## 5. Error Estimate

In this section, we provide in theL2 norm an estimate of the gap between the approximate solution and the exact solution.Lemma 7.
Assume thatHρ0, HG, Hq.p., and HV1 hold. Let ρ=ρt,x be a weak solution of the q(x)-Laplacian parabolic equation and ρh=ρht,x be an approximate solution of the q(x)-Laplacian parabolic equation defined in (12). Then,(153)ρ−ρhL20,T×Ω2≤CT,N,M,ρ,Ω,G,p.,q.h,where CT,N,M,ρ,Ω,G,p.,q. is a constant depending on T,N,M,ρ,Ω,G,p.,q..Proof 10.
Sinceρ=ρt,x is a weak solution of the q(x)-Laplacian parabolic equation, then(154)ρt,x=ρk−1x+∫hk−1tdivxρ∇xG′ρ+Vqx−2∇xG′ρ+Vds,fort∈hk−1,hk.
Similarly, the approximate solutionρh satisfies(155)ρht,x=ρk−1x+∫hk−1tdivxρh∇xG′ρh+Vqx−2∇xG′ρh+Vds,fort∈hk−1,hk.
We combine (154) and (155) to have(156)ρt,x−ρht,x=∫hk−1tdivxρ∇xG′ρ+Vqx−2∇xG′ρ+V−divxρh∇xG′ρh+Vqx−2∇xG′ρh+Vds.
We multiply (156) by G′ρ−G′ρh, and we integrate it on 0,T×Ω. Consequently,(157)∫0,T×Ωρt,x−ρht,xG′ρ−G′ρhdtdx=−∫0,T×Ω∫hk−1t<ρσqx−2σ−ρhσhqx−2σh,σ−σh>dtdxds,where σ=∇xG′ρ+V and σh=∇xG′ρh+V.
SinceN≤ρ≤M, N≤ρh≤M, and G is a convex function of class C2, then G′′≥0, and(158)ρt,x−ρht,xG′ρ−G′ρh=G′′ρh+θρ−ρhρ−ρh2≥k0ρ−ρh2,where k0=inft∈N;MG′′t and θ∈0;1. Then, we combine (157) and (158) to deduce that(159)∫0,T×Ωρt,x−ρht,x2dtdx≤−1k0∫0,T×Ω∫hk−1t<ρσqx−2σ−ρhσhqx−2σh,σ−σh>dtdxds.
SinceN≤ρ≤M and N≤ρh≤M, then(160)−1k0∫0,T×Ω∫hk−1t<ρσqx−2σ−ρhσhqx−2σh,σ−σh>dtdxds≤2Mk0hσqx−2σLp.0,T×Ω+σhqx−2σhLp.0,T×ΩσLq.0,T×Ω+σhLq.0,T×Ω.
By usingσ=∇xG′ρ+V and σh=∇xG′ρh+V, we obtain(161)σhLq.0,T×Ω≤max∫0,T×Ω∇xG′ρh+Vqxdtdx1/q+,∫0,T×Ω∇xG′ρh+Vqxdtdx1/q−,(162)σhqx−2σhLp.0,T×Ω≤max∫0,T×Ω∇xG′ρh+Vqxdxdt1/p+,∫0,T×Ω∇xG′ρh+Vqxdtdx1/p−.
By using (64), then(163)∫0,T×Ω∇xG′ρh+Vqxdtdx≤Tp+N∫ΩGρ0dx−ΩG1Ω+2VL∞Ω.
Finally, we combine (159)–(163) to obtain the desired estimate:(164)∫0,T×Ωρt,x−ρht,x2dtdx≤CT,N,M,Ω,G,p.,q.h,where CT,N,M,Ω,G,p.,q. is a constant depending on T,N,M,Ω,G,p.,q..


## 6. Numerical Example

Here, we give an illustration of our work by testing the algorithm on an example of theq(x)-Laplacian parabolic equation in the dimension equal to 1 for determining an approximate solution. We limit ourselves to the first-order approximation just for the sake of simplicity and to avoid cumbersome calculations. An error table is obtained from different values of step h, showing that the error values decrease to zero in the L2−norm when h tends to zero, in relevance with the similarity between the graphical representations of both the analytical solution and the approximate solution.q(x)-Laplacian parabolic equation is a nonhomogeneous equation and may be considered as a model of response over the time of nonhomogeneous media under the stress of a physical phenomenon. We suppose in our example that the functions p,V, and G in the q(x)-Laplacian parabolic equation are as follows:(165)Vx=x22χ1/2,3/4,ifx∈0,1,qx=52χ0,1/2∪3/4,1+2χ1/2,3/4,Gx=94x34χ0,1/2∪3/4,1+xlnxχ1/2,3/4,forx∈0,1.Consequently, the parabolicq(x)-Laplacian parabolic equation becomes the following couple of the linear Fokker–Planck equation and the 5/2-Laplacian parabolic equation:(166)∂ρ∂t=Δxρ+divxxρin0,+∞×12,34,∂ρ∂t=divx∇xρ1/2∇xρin0,+∞×0,12∪0,+∞×34,1.One can assimilate the set0,1 to a nonhomogeneous composite rod of length 1 cast in two distinct materials and made of a main piece 1/2,3/4 surrounded at its extremities by two other pieces 0,1/2 and 3/4,1. The nonhomogeneity is characterized by the two different constant values taken by the variable exponent qx on the distinct pieces of the rod.Let us express the solution of the linear Fokker–Planck equation(167)∂ρ∂t=Δxρ+divxxρ,in the form(168)ρt,x=etuτ,y,τ,y=12e2t,etx,where uτ,y is an unknown function, supposed to be a solution of the heat equation(169)∂u∂τ=Δyu.Thus, after a simple computation, we have that(170)∂ρ∂t=etuτ,y+e3t∂u∂τ+xe2t∂u∂y,Δxρ=e3tΔyu,divxxρ=etuτ,y+xe2t∂u∂y.So, ifu=ut,x satisfies(171)∂u∂τ=Δyu,then ρt,x=etuτ,y satisfies(172)∂ρ∂t=Δxρ+divxxρ.Besides, the functionuτ,y=1/4τ+1e−y2/4τ+1 is an exact solution of heat equation (169). Accordingly, the function ρt,x=etu1/2e2t,etx defined by(173)ρt,x=et2e2t+1e−e2tx2/2e2t+1,is an exact solution of equation (167).Next, let us deal with the parabolic5/2-Laplacian equation:(174)∂ρ∂t=divx∇xρ1/2∇xρ.We obtain an exact solutionρ=ρt,x of (174) in the form(175)ρt,x=4x5530t+12.Finally, fix the initial datumρ0 to be(176)ρ0x=45x5,ifx∈0,12∪34,1,ρ0x=13e−x2/3,ifx∈12,34.Then, the exact solutionρ=ρt,x of the q(x)-Laplacian parabolic equation is presented as follows:(177)ρt,x=4x5530t+12,ift,x∈0,+∞×0,12∪0,+∞×34,1,ρt,x=et2e2t+1e−e2tx2/2e2t+1,ift,x∈0,+∞×12,34.In this part of our work, we are interested in the existence of the stationary solution for our example. With the functionsG and V as defined in the above, we deduce that the equation(178)ρ∞∇xG′ρ∞+V=0becomes(179)ρ∞∇xρ∞1/3=0,in0,12∪34,1,ρ∞∇x1+lnρ∞+x22=0,in12,34.Consequently, the functionρ∞ which is defined by(180)ρ∞x=0,ifx∈0,12∪34,1,ρ∞=12e−x2/2,ifx∈12,34solves (179) as the stationary solution of the q(x)-Laplacian parabolic equation.In order to appreciate the effectiveness of our approximation method, we apply descent algorithm (6) to derive the approximate solution ρh for comparison with the exact solution, and we get(181)ρht,x=451+50hx5,ift,x∈0,h×0,12∪0,h×34,1,ρht,x=33−he−x2/3,ift,x∈0,h×12,34.


### 6.1. The Figures

For some values ofh, we have drawn the figures of both analytical and approximate solutions when step h∈1/10,1/50,1/100, using Scilab software. We mention that Figure 1 represents an approximation of Figure 2 when, Figure 3 represents an approximation of Figure 4 when h=1/100, and Figure 5 represents an approximation of Figure 6 when h=1/1000.Figure 1
Numerical solutionρ11/10x for h=1/10.Figure 2
Analytical solutionρ1/20,x for h=1/10 and t=1/20∈0,1/10.Figure 3
Numerical solutionρ1/1000x for h=1/100.Figure 4
Analytical solutionρ1/200,x for h=1/100 and t=1/2000∈0,1/1000.Figure 5
Numerical solutionρ1/1000x for h=1/1000.Figure 6
Analytical solutionρ1/2000,x for h=1/1000 and t=1/2000∈0,1/1000.


### 6.2. Table of Error Progression

The Table of Error Progression is provide in the Table1Table 1
Error table.
h1/101/501/1001/5001/10001/50001/10000δt1/201/1001/2001/10001/20001/200001/20000ρ−ρhL214.1443433.38861301.78476950.37641190.18963130.03816220.0190961


## 6.1. The Figures

For some values ofh, we have drawn the figures of both analytical and approximate solutions when step h∈1/10,1/50,1/100, using Scilab software. We mention that Figure 1 represents an approximation of Figure 2 when, Figure 3 represents an approximation of Figure 4 when h=1/100, and Figure 5 represents an approximation of Figure 6 when h=1/1000.Figure 1
Numerical solutionρ11/10x for h=1/10.Figure 2
Analytical solutionρ1/20,x for h=1/10 and t=1/20∈0,1/10.Figure 3
Numerical solutionρ1/1000x for h=1/100.Figure 4
Analytical solutionρ1/200,x for h=1/100 and t=1/2000∈0,1/1000.Figure 5
Numerical solutionρ1/1000x for h=1/1000.Figure 6
Analytical solutionρ1/2000,x for h=1/1000 and t=1/2000∈0,1/1000.


## 6.2. Table of Error Progression

The Table of Error Progression is provide in the Table1Table 1
Error table.
h1/101/501/1001/5001/10001/50001/10000δt1/201/1001/2001/10001/20001/200001/20000ρ−ρhL214.1443433.38861301.78476950.37641190.18963130.03816220.0190961



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*Source: 9756162-2020-07-25.xml*