Existence of Positive Solutions and Asymptotic Behavior for Evolutionary q(x)-Laplacian Equations

<jats:p>In this paper, we extend the variational method of M. Agueh to a large class of parabolic equations involving <jats:italic>q</jats:italic>(<jats:italic>x</jats:italic>)-Laplacian parabolic equation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mo>∂</mml:mo><mml:mi>ρ</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>x</mml:mi></mml:mrow></mml:mfenced><mml:mo>/</mml:mo><mml:mrow><mml:mo>∂</mml:mo><mml:mi>t</mml:mi></mml:mrow></mml:mrow></mml:mfenced><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mtext>div</mml:mtext></mml:mrow><mml:mrow><mml:mi>x</mml:mi></mml:mrow></mml:msub><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mfenced open="" close="|" separators="|"><mml:mrow><mml:mi>ρ</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>x</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:mfenced><mml:msub><mml:mrow><mml:mo>∇</mml:mo></mml:mrow><mml:mrow><mml:mi>x</mml:mi></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mfenced open="" close="|" separators="|"><mml:mrow><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:msup><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow></mml:mfenced><mml:mo>+</mml:mo><mml:mi>V</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:mfenced></mml:mrow><mml:mrow><mml:mi>q</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>x</mml:mi></mml:mrow></mml:mfenced><mml:mo>−</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:msub><mml:mrow><mml:mo>∇</mml:mo></mml:mrow><mml:mrow><mml:mi>x</mml:mi></mml:mrow></mml:msub><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:msup><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mrow><mml:mo>′</mml:mo></mml:mrow></mml:msup><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow></mml:mfenced><mml:mo>+</mml:mo><mml:mi>V</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:mfenced></mml:math>. The potential <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mi>V</mml:mi></mml:math> is not necessarily smooth but belongs to a Sobolev space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:msup><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:msup><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi mathvariant="normal">Ω</mml:mi></mml:mrow></mml:mfenced></mml:math>. Given the initial datum <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:msub><mml:mrow><mml:mi>ρ</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math> as a probability density on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mi mathvariant="normal">Ω</mml:mi></mml:math>, we use a descent algorithm in the probability space to discretize the <jats:italic>q</jats:italic>(<jats:italic>x</jats:italic>)-Laplacian parabolic equation in time. Then, we use compact embedding <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:msup><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mfenced></mml:mrow></mml:msup><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi mathvariant="normal">Ω</mml:mi></mml:mrow></mml:mfenced></mml:math>↪↪<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M7"><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>q</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mo>.</mml:mo></mml:mrow></mml:mfenced></mml:mrow></mml:msup><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi mathvariant="normal">Ω</mml:mi></mml:mrow></mml:mfenced></mml:math> established by Fan and Zhao to study the convergence of our algorithm to a weak solution of the <jats:italic>q</jats:italic>(<jats:italic>x</jats:italic>)-Laplacian parabolic equation. Finally, we establish the convergence of solutions of the <jats:italic>q</jats:italic>(<jats:italic>x</jats:italic>)-Laplacian parabolic equation to equilibrium in the <jats:italic>p</jats:italic>(.)-variable exponent Wasserstein space.</jats:p>


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 the q (x)-Laplacian operator: where Ω ⊂ R d is a convex, bounded, and smooth domain of R d , G: [0, +∞[ ⟶ R is a convex function of class C 2 (]0, ∞[), q(.): Ω ⟶ ]1, +∞] is a continuous function, and V: Ω ⟶ R belongs to a Sobolev space W 1,∞ (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 the q(x)-Laplacian parabolic equation with the variable exponent, when G(t) � t 2 and V ≡ 0 (see [1,2,6]), for a given initial datum ρ 0 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: ρ 0 ∈ L ∞ (Ω) and |∇ x ρ 0 | q + ∈ L 1 (Ω).
In [7], M. Agueh studied the existence of positive solutions for the q(x)-Laplacian parabolic equation when the variable exponent q(x) ≡ q is constant (with q > 1), and the potential V is a convex function of class C 1 (Ω). Moreover, the author in [7] proved that the parabolic q-Laplacian equation is a gradient flow of the functional E(ρ) � Ω (G(ρ) + ρV)dx with respect to the p-Wasserstein distance (p � (q/q − 1)) defined by where ρ 1 and ρ 2 are two probability densities on Ω.
In fact, by fixing the time step h > 0 and a probability density ρ 0 on Ω, the author defines ρ k− 1 (k ∈ N * ) as the solution of (2) at t � h(k − 1) and ρ k as the solution of (2) at t � hk such that ρ k is the unique solution of the variational problem en, the author established the convergence of the approximate solutions (ρ k ) k 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 W 1,∞ (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 E(ρ) � Ω [G(ρ) + ρV]dx with respect to the p(.)− Wasserstein distance W p(.) defined by (t,.) ‖v‖ L p (.) ρ(t,.) ([0,1]×Ω) , Next, we proceed with the discretization of the q(x)-Laplacian parabolic equation as follows: fixing the time step h > 0 and a probability density ρ 0 on Ω, we define ρ k as the approximate solution of the q(x)-Laplacian parabolic equation at t k � hk, which minimizes the variational problem Here, Π(ρ k− 1 , ρ) is the set of all probability measures on Ω × Ω having ρ k− 1 and ρ as their marginals, and is the Monge-Kantorovich work associated to the cost c h (x, y) � (|x − y| p(x) /p(x)h p(x)− 1 ). e establishment of our result will be derived according to the following steps: (1) Given ρ 0 as a probability density on Ω such as N ≤ ρ 0 ≤ M and V ∈ L ∞ (Ω), we prove that (P k ) admits a unique solution ρ k which satisfies N ≤ ρ 0 ≤ M (see Lemma 1). (2) We prove in (28) that the Kantorovich problem admits a unique solution c k in Π(ρ k− 1 , ρ k ) and that suppc k satisfy where q(x) � (p(x)/p(x) − 1). Here, V ∈ W 1,∞ (Ω) being not necessarily smooth, we approximate V by C 2 (Ω)-functions, and we use descent algorithm (6), the maximum principle N ≤ ρ k ≤ M, and compact embedding W 1,q(x) (Ω)↪↪L q(x) (Ω) to establish (10).
(3) We now use the maximum principle N ≤ ρ k ≤ M and (10) to prove that the sequence (ρ k ) k is a time discretization of the nonlinear q(x)-Laplacian parabolic equation, that is, for all test functions, ϕ ∈ C ∞ c (R d ), Afterward, we establish the following: ′ To prove (i), we use descent algorithm (6) and the maximum principle N ≤ ρ h ≤ M to deduce that the sequence Fan and Zhao in [9], we conclude that the sequence (ρ h ) h converges strongly to We now use (i) and the fact that v ⟼ |v| q(x) is convex for all x ∈ Ω fixed to establish (ii). We combine (i) and (ii) to prove that the sequence (ρ h ) h 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 the q(x)-Laplacian parabolic 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 exponent q(.): 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: (13) en, the sequence (ρ h ) h converges to a weak solution ρ(t, x) of the q(x)-Laplacian parabolic equation. en, where t 1 ≔ infI 1 and t 2 ≔ infI 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.

Existence of Solutions for the Nonlinear q(x)-Laplacian Parabolic Equation
In this section, we prove the existence of solutions for the nonlinear q(x)-Laplacian parabolic equation.

Euler-Lagrangian Equation for Variational Problem (37).
Here, we show that the sequence (ρ k ) k defined in (37) is a time discretization of the q(x)-Laplacian parabolic equation, i.e., for all test functions, where A h k (ϕ) converges to 0 when h tends to 0. where of the maximum principle is carried out similarly as given in [7].
Let ρ ∈ P(Ω); since G is convex and V ∈ L ∞ (Ω), G being positive and convex, then Since the variable exponent p(.) is continuous on Ω, the Kantorovich problem admits a solution c n . Moreover, since Ω is bounded, the sequence (c n ) n converges to a measure c 1 in P(Ω × Ω) narrowly, and c 1 ∈ Π(ρ 0 , ρ 1 ); and then, we derive that We combine (35) and (33) to obtain lim inf us, ρ 1 is a solution of variational problem (P 1 ). From the strict convexity of G, we deduce that ρ ⟼ Ω (G(ρ) + ρV)dx is strictly convex and so is ρ ⟼ I(ρ) on P(Ω), and consequently, the uniqueness of the solution ρ 1 of (P 1 ) follows.
□ Now, we assume that (H ρ 0 ), (H G ), and (H 1 V ) hold. en, from Lemma 1, we obtain that the variational problem admits a unique solution ρ k for all k ≥ 1.
Next, we prove that (ρ k ) k is a time discretization of the nonlinear q(x)-Laplacian parabolic equation. In order to achieve this, we use the following lemma.
admits a solution c k such that

Proof 2.
e proof of Lemma 2 is derived following the two steps.

□
Step 1. We first assume that V ∈ C 2 (Ω). Fix Define the probability density ρ ϵ as ρ ϵ � T ϵ #ρ k and c ϵ as a probability measure on Ω × Ω defined by Since G satisfies (H G ), then Discrete Dynamics in Nature and Society 5 Since V ∈ C 2 (Ω), from the Taylor formula, we have (see [7]) en, we have after integration, By using (41) and the fact that We now use the dominated convergence theorem to have Note that c ϵ defined in (42) belongs to Π(ρ k− 1 , ρ ϵ ), and So, for ϵ > 0, we have Note that Indeed, (ii) On the contrary, the Taylor formula with respect to ϵ enables us to write where θ ∈ (0, 1) and T ϵ (y) � y + ϵψ(y). Also, we have We then combine the results given in (i) and (ii) and the dominated convergence theorem to obtain (50).
By fixing δ > 0, we define the sequence (ρ δ k ) k such that ρ δ 0 � ρ 0 with ρ δ k (for k ≥ 1), the solution of the variational problem is defined as in (38). As in Lemma 1, the variational problem (P δ k ) admits a unique solution ρ δ k in P(Ω), and N ≤ ρ δ k ≤ M. Hence, the Kantorovich problem admits a solution c δ k such that Let us show that (ρ δ k ) δ converges to ρ k and (c δ k ) δ converges to c k up to a subsequence, as well as suppc k satisfies (40) However, G is convex, and V ∈ W 1,∞ (Ω); and then, Furthermore, recalling (63) and the fact that is compact (see [9]), and hence, the sequence (ρ δ k ) δ converges strongly to some u k in L q(x) (Ω) up to a subsequence. Moreover, ρ δ k minimizes (P δ k ) for k ∈ N * , and then, we have for all ρ ∈ P(Ω). Also, from the boundedness of Ω, (c δ k ) δ converges narrowly to a measure Γ k in P(Ω × Ω), and Discrete Dynamics in Nature and Society Γ k ∈ Π(u k− 1 , u k ). en, we use the strong convergence of (ρ δ k ) δ to u k and the fact that ‖V δ − V‖ W 1,∞ (Ω) converges to 0 when δ ⟶ 0 to obtain that for all ρ ∈ P(Ω). Since ρ δ 0 � ρ 0 , we conclude that u k � ρ k and Γ k � c k for all k ∈ N * . By using (63) and N ≤ ρ k ≤ M, we deduce that Next, we use (57), and we have and (c δ k ) δ converges narrowly to c k in P(Ω × Ω); then, tending δ to 0 in (67), we obtain for all test functions, ψ ∈ C ∞ c (R d , R d ). Finally, we obtain the equality Now, let us prove that (ρ k ) k is a time discretization of a nonlinear q(x)-Laplacian parabolic equation. Let We now use (69) and the Taylor formula to obtain

Let us show that
converge to 0 when h tends to 0. Since Define We use (64) and (73) We tend h to 0, and we conclude that A h k (ψ) converges to 0. us, the sequence (ρ k ) k is a time discretization of the nonlinear q(x)-Laplacian parabolic equation. Define We prove in the next section that the sequence (ρ h ) h converges strongly to a function ρ(t, x) in L q(x) ([0, T] × Ω) for 0 < T < ∞ and that the nonlinear term sequence div

Strong
Convergence of (ρ h ) h and Weak Convergence of the Nonlinear Term Sequence. Here, we denote the nonlinear term sequence by div us, we use (64), and we have where 0 < T < ∞. We conclude that (ρ h (t, .)) h is bounded in en, up to a subsequence, the sequence (ρ h ) h converges strongly to ρ(t, x) in L q(x) ([0, T] × Ω). By using (64) and en, since G ′ is continuous and us, the sequence (ω h ) h converges weakly to some ω ∈ L p(x) ([0, T] × Ω).
As in [7], we derive easily that div e proof of this lemma is achieved via the following three claims: Discrete Dynamics in Nature and Society Using (81), we have lim inf We now use (83) and (84) in (82) to conclude the proof of claim 1.
(86) □ Proof 5. We use the descent algorithm (P k ) and (70) to obtain (87) en, we obtain after integration over [0, T] that 1 h We use p(x) ≤ p + and τ � (h/p + ) in (87), and we have where ρ τ is defined as follows: and Using the fact that suppu ⊂ [− T, T], we have Recalling inequalities (88) and (89), we get We tend τ toward 0, and we obtain lim sup From the strict convexity of G ∈ C 1 (]0, ∞[), we derive that where G * is a Legendre transform of G.
We conclude the proof of claim 2 by using (92) in (93). Claim 3: Discrete Dynamics in Nature and Society 11 where o(h) converges to 0 when h tends to 0. As in (89), we have We tend h to 0 in (98) and use the fact that ρ h converges strongly to ρ in L q(x) ([0, T] × Ω) and ω h weakly to ω in By using we have Next, after integration of the inequality above, we have Note that Using the change of variable s � t − h in (105), we obtain We combine (106) and (105) to deduce that Now, we use (107) in (108) to derive So, (103) becomes Discrete Dynamics in Nature and Society We deduce that Now, we combine (99) and (111) to obtain Finally, we conclude that is completes the proof of claim 3. Now, we are ready to show that the sequence So, we obtain 14 Discrete Dynamics in Nature and Society We use Lemma 3 and we tend h to 0 in (115), and then we get So, Dividing (116) by ϵ > 0 and tending ϵ to 0, we reach By replacing ψ by − ψ, we have us, we conclude that (div x (ρ h ω h )) h converges weakly to div

Proof of the eorem of the Laplacian Parabolic Equation.
Here, we use the strong convergence of the sequence (ρ h ) h to ρ in L q(x) ([0, T] × Ω) and the weak convergence of the nonlinear term where o(h) tends to 0 when h tends to 0. Note that en, (120) becomes Finally, we tend h to 0 in (122), and we use the fact that ρ h converges strongly to ρ and that (div We deduce that Discrete Dynamics in Nature and Society It follows that ρ(t, x) is a weak solution of a nonlinear q(x)-Laplacian parabolic equation.

Asymptotic Behavior
In this section, we give the proof of eorem 2 which is derived from the following three lemmas. Lemma 4. Let ρ 1 and ρ 2 be two probability densities on Ω. In for all ρ ∈ P(Ω).
Next, we have after integration, on I 2 , and on I 1 .
To conclude, we use Lemma 4 and expressions (151) and (152), and then, the proof of eorem 2 is complete.

Error Estimate
In this section, we provide in the L 2 norm an estimate of the gap between the approximate solution and the exact solution. (12). en,

) be a weak solution of the q(x)-Laplacian parabolic equation and ρ h � ρ h (t, x) be an approximate solution of the q(x)-Laplacian parabolic equation defined in
where C (T, N Similarly, the approximate solution ρ h satisfies We combine (154) and (155) to have en, the exact solution ρ � ρ(t, x) of the q(x)-Laplacian parabolic equation is presented as follows: In this part of our work, we are interested in the existence of the stationary solution for our example. With the functions G and V as defined in the above, we deduce that the equation Consequently, the function ρ ∞ which is defined by   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).

Table of Error Progression.
e Table of Error Progression is provide in the Table 1 Data Availability e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.