Analysis, Control and Applications of Passivity in Complex NetworksView this Special Issue
Existence of Positive Solutions and Asymptotic Behavior for Evolutionary q(x)-Laplacian Equations
In this paper, we extend the variational method of M. Agueh to a large class of parabolic equations involving q(x)-Laplacian parabolic equation . The potential is not necessarily smooth but belongs to a Sobolev space . Given the initial datum as a probability density on , we use a descent algorithm in the probability space to discretize the q(x)-Laplacian parabolic equation in time. Then, we use compact embedding ↪↪ 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.
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 the q (x)-Laplacian operator:where is a convex, bounded, and smooth domain of , is a convex function of class , is a continuous function, and belongs to a Sobolev space .
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 , 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 and (see [1, 2, 6]), for a given initial datum 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: and .
In , M. Agueh studied the existence of positive solutions for the q(x)-Laplacian parabolic equation when the variable exponent is constant (with ), and the potential is a convex function of class . Moreover, the author in  proved that the parabolic q-Laplacian equationis a gradient flow of the functional with respect to the p-Wasserstein distance () defined bywhere and are two probability densities on .
In fact, by fixing the time step and a probability density on , the author defines () as the solution of (2) at and as the solution of (2) at such that is the unique solution of the variational problem
Then, the author established the convergence of the approximate solutions to a weak solution of the Laplacian parabolic equation . Here, we extend the work of  to a general case where may not be smooth but belongs to a Sobolev space . 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 , we prove that the Laplacian parabolic equation is a gradient flow of the functional with respect to the Wasserstein distance defined by
Next, we proceed with the discretization of the q(x)-Laplacian parabolic equation as follows: fixing the time step and a probability density on , we define as the approximate solution of the q(x)-Laplacian parabolic equation at , which minimizes the variational problemwhere
Here, is the set of all probability measures on having and as their marginals, andis the Monge–Kantorovich work associated to the cost .
The establishment of our result will be derived according to the following steps:(1)Given as a probability density on such as and , we prove that admits a unique solution which satisfies (see Lemma 1).(2)We prove in (28) that the Kantorovich problem admits a unique solution in and that satisfy where . Here, being not necessarily smooth, we approximate by -functions, and we use descent algorithm (6), the maximum principle , and compact embedding ↪↪ to establish (10).(3)We now use the maximum principle and (10) to prove that the sequence is a time discretization of the nonlinear q(x)-Laplacian parabolic equation, that is, for all test functions, , where converges to 0 when tends to 0.(4)We define by
Afterward, we establish the following:(i)The strong convergence of the sequence to a function in for (ii)The weak convergence of nonlinear term to in
To prove (i), we use descent algorithm (6) and the maximum principle to deduce that the sequence is bounded in for . Then, taking into account the compact embedding ↪↪ established by Fan and Zhao in , we conclude that the sequence converges strongly to in .
We now use (i) and the fact that is convex for all fixed to establish (ii). We combine (i) and (ii) to prove that the sequence 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 equation to when , where is the probability density which satisfies .
Note that in [10, 11], the authors proved a convergence of solutions to the equilibrium without specifying the speed of convergence. In , the long-time behavior of solutions of the q(x)-Laplacian parabolic equation is only established if .
In this paper, we extend to the variable exponent such that , 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 that , , , and hold, and define by
Then, the sequence converges to a weak solution of the q(x)-Laplacian parabolic equation.
Theorem 2. Assume that , , , , and hold. Let be a weak solution of the q(x)-Laplacian parabolic equation. Define
Then,where and .
Throughout this work, we will assume the following: is a probability density on , with , for some , and . : are continuous functions such that for all , and ; . is strictly convex, , and is convex. is a potential which satisfies , , and
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. Let be a probability measure on and be a measurable function. We denote by the Lebesgue space with the variable exponent defined bywith the normfor all .
We denote by the Sobolev space with the variable exponent defined byequipped with the normIn particular, if , then and .
We also define the space as the closure of the space of -functions with compact support in in the space endowed with the normIt is well known (cf. ) that and are Banach spaces, respectively, with norms (20) and (22).
Proposition 1 (Hölder inequality; see ). Let and be two measurable functions such that , for all .
If and , then
Furthermore, if are measurable functions such that , for almost all , we havefor and .
Proposition 2 (see ). Let and be two measurable functions such that on . Then, we have the following continuous injection:
Furthermore,for all .
Proposition 3 (see ). Assume that holds. Then, the following statements hold:(i)The Banach spaces and are separable and reflexive Banach spaces(ii)The embedding ↪↪ is continuous and compact(iii)There is a constant such that for all
3. 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.
3.1. Euler–Lagrangian Equation for Variational Problem (37)
Here, we show that the sequence defined in (37) is a time discretization of the q(x)-Laplacian parabolic equation, i.e., for all test functions, , we havewhere converges to 0 when tends to 0.
Lemma 1. Assume that , , and hold. Then, the variational problemwhereadmits a unique solution in which satisfies .
Proof 1. Since , then the solution of variational problem (if there exists) satisfies . The proof of the maximum principle is carried out similarly as given in .
Since and , we have .
Let ; since is convex and ,We conclude that is finite.
Let be a sequence in such that and converges to . Then, converges weakly∗ to in up to a subsequence.
being positive and convex, thenNext, we use the fact that and to obtainSince the variable exponent is continuous on , the Kantorovich problemadmits a solution . Moreover, since is bounded, the sequence converges to a measure in narrowly, and ; and then, we derive thatWe combine (35) and (33) to obtainThus, is a solution of variational problem . From the strict convexity of , we deduce that is strictly convex and so is on , and consequently, the uniqueness of the solution of follows.
Now, we assume that , , and hold. Then, from Lemma 1, we obtain that the variational problemwithadmits a unique solution for all .
Next, we prove that 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 that , , and hold. Then, the Kantorovich problemadmits a solution such thatwhere .
Proof 2. The proof of Lemma 2 is derived following the two steps.
Step 1. We first assume that .
Fix . Let in defined byDefine the probability density as and as a probability measure on defined byfor all .
For ,Since satisfies , thenSince , from the Taylor formula, we have (see )Then, we have after integration,By using (41) and the fact that , we haveWe now use the dominated convergence theorem to haveNote that defined in (42) belongs to , andSo, for , we haveNote thatIndeed,(i)We have(ii)On the contrary, the Taylor formula with respect to enables us to writewhere and . Also, we havewithWe then combine the results given in (i) and (ii) and the dominated convergence theorem to obtain (50).
Since minimizes on , thenSo, we combine (44), (48), and (50) to obtain thatReplacing by in (56), we getFrom (56) and (57), we deduce thatThus, we obtain (40) when .
Now, let us establish the proof of the lemma when and is nonregular.
Step 2. Assume that . Let be a sequence in such that .
By fixing , we define the sequence such that with (for ), the solution of the variational problemwhere is defined as in (38). As in Lemma 1, the variational problem admits a unique solution in , and . Hence, the Kantorovich problemadmits a solution such thatLet us show that converges to and converges to up to a subsequence, as well as satisfies (40)
Using and (61), we haveHowever, is convex, and ; and then,Furthermore, recalling (63) and the fact that , , and , we obtainwhere . We conclude that is bounded on . Since is continuous and is a bounded set, the injection ↪↪ is compact (see ), and hence, the sequence converges strongly to some in up to a subsequence. Moreover, minimizes for