Abstract
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 [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 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 [7], 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 [7] 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 [7] 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 [7], 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 [9], 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 [11], 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 .
2. Preliminaries
2.1. Assumptions
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. [9]) that and are Banach spaces, respectively, with norms (20) and (22).
Proposition 1 (Hölder inequality; see [12]). 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 [9]). Let and be two measurable functions such that on . Then, we have the following continuous injection:
Furthermore,for all .
Proposition 3 (see [9]). 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 [7].
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 [7])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 [9]), and hence, the sequence converges strongly to some in up to a subsequence. Moreover, minimizes for , and then, we havefor all . Also, from the boundedness of , converges narrowly to a measure in , and . Then, we use the strong convergence of to and the fact that converges to 0 when to obtain thatfor all . Since , we conclude that and for all .
By using (63) and , we deduce that is bounded in . Since converges strongly to in and is continuous, then converges weakly to in .
Next, we use (57), and we haveSince converges strongly to in , converges weakly to in , and converges narrowly to in ; then, tending to 0 in (67), we obtainfor all test functions, .
Finally, we obtain the equalityNow, let us prove that is a time discretization of a nonlinear q(x)-Laplacian parabolic equation. Let be a test function; we haveWe now use (69) and the Taylor formula to obtainwhere and is the transpose of .
Let us show thatconverge to 0 when tends to 0. Since , we haveDefine and :We use (64) and (73) in (72), and we haveWe tend to 0, and we conclude that converges to 0. Thus, the sequence is a time discretization of the nonlinear q(x)-Laplacian parabolic equation.
Define as follows:We prove in the next section that the sequence converges strongly to a function in for and that the nonlinear term sequence converges weakly to in .
3.2. Strong Convergence of and Weak Convergence of the Nonlinear Term Sequence
Here, we denote the nonlinear term sequence by , with .
Since , then . Thus, we use (64), and we havewhere . We conclude that is bounded in for almost all . By using the compact injection ↪↪, we deduce that converges strongly to a function in . Since and is bounded, we use the dominated convergence theorem to have
Then, up to a subsequence, the sequence converges strongly to in . By using (64) and , we deduce that is bounded in . Then, since is continuous and converges strongly to in , we conclude that converges weakly to in .
Note that . Then, using the fact that is bounded in , we conclude that is bounded in . Thus, the sequence converges weakly to some .
As in [7], we derive easily that converges weakly to in .
Lemma 3. Fix . Let be a test function such that . We have
Proof 3. The proof of this lemma is achieved via the following three claims:
Claim 1:
Proof 4. Since , , and is monotone for all fixed in , we haveUsing (81), we haveSince converges strongly to in , converges weakly to in , and converges weakly to in , thenWe now use (83) and (84) in (82) to conclude the proof of claim 1.
Claim 2:
Proof 5. We use the descent algorithm and (70) to obtainNote that , with . Then, we obtain after integration over thatWe use and in (87), and we havewhere is defined as follows:and .
Using the fact that , we haveRecalling inequalities (88) and (89), we getWe tend toward 0, and we obtainFrom the strict convexity of , we derive thatwhere is a Legendre transform of .
We conclude the proof of claim 2 by using (92) in (93).
Claim 3:
Proof 6. Define , for .
, and then we approximate by -functions, and (70) becomeswhere converges to 0 when tends to 0. As in (89), we havewhere .
Using (96), (97) becomesWe tend to 0 in (98) and use the fact that converges strongly to in and weakly to in to getBy usingwe haveNext, after integration of the inequality above, we haveNote thatUsing the change of variable in (105), we obtainWe combine (106) and (105) to deduce thatNow, we use (107) in (108) to deriveSo, (103) becomesSince , thenWe deduce thatNow, we combine (99) and (111) to obtainFinally, we conclude thatThis completes the proof of claim 3.
Now, we are ready to show that the sequence converges weakly to in . Let and be some positive functions such that , with . For , we define , and thenSo, we obtainWe use Lemma 3 and we tend to 0 in (115), and then we getSo,Dividing (116) by and tending to 0, we reachBy replacing by , we havefor all test functions and such that .
Thus, we conclude that converges weakly to in .
3.3. Proof of the Theorem of the Laplacian Parabolic Equation
Here, we use the strong convergence of the sequence to in and the weak convergence of the nonlinear term to in , and we establish that is a weak solution of the q(x)-Laplacian parabolic equation. Let be a test function such that . By using (71), we obtainwhere tends to 0 when tends to 0.
Note that
Then, (120) becomes
Finally, we tend to 0 in (122), and we use the fact that converges strongly to and that converges weakly to :
We deduce that
It follows that is a weak solution of a nonlinear q(x)-Laplacian parabolic equation.
4. Asymptotic Behavior
In this section, we give the proof of Theorem 2 which is derived from the following three lemmas.
Lemma 4. Let be two probability densities on . Assume that satisfies . If is the solution of the Monge problemthen
In particular, if satisfies , thenfor all .
Here, is the -Wasserstein distance defined in (5).
Proof 7. Let . Fix . It is known from the work in [8] that the Monge problemadmits a unique solution . Therefore, since and satisfy, respectively, and , thenwhere and is the -Wasserstein distance.
The continuous injection ↪ implies thatSo, we use (129) and (128) to deriveIn particular, if satisfies and , thenSo, we havefor all .
Note that is a stationary solution of the q(x)-Laplacian parabolic equation, which minimizes the functional on .
Lemma 5. Let . Then,with .
Proof 8. We applied Young’s inequality to obtainwhere and . Then,and since , we use the continuous injection ↪, and we getNow, we combine (128), (137), and (136) to obtainIn particular, if , we obtain (135).
Lemma 6.
Proof 9. From [9], we getandSince , we haveBy using (141), we derive thatNext, we combine (143) and (144) to deriveBy using (142) and (143), we obtain (140).
4.1. Proof of Theorem 2
Let be a solution of the q(x)-Laplacian parabolic equation. We have
Define
We combine (135), (142), and (146) to geton , and we combine (135), (145), and (146) to haveon .
Next, we have after integration,on , andon .
To conclude, we use Lemma 4 and expressions (151) and (152), and then, the proof of Theorem 2 is complete.
5. Error Estimate
In this section, we provide in the norm an estimate of the gap between the approximate solution and the exact solution.
Lemma 7. Assume that , , , and hold. Let be a weak solution of the q(x)-Laplacian parabolic equation and be an approximate solution of the q(x)-Laplacian parabolic equation defined in (12). Then,where is a constant depending on .
Proof 10. Since is a weak solution of the q(x)-Laplacian parabolic equation, thenSimilarly, the approximate solution satisfiesWe combine (154) and (155) to haveWe multiply (156) by , and we integrate it on . Consequently,where and .
Since , , and is a convex function of class , then , andwhere and . Then, we combine (157) and (158) to deduce thatSince and , thenBy using and , we obtainBy using (64), thenFinally, we combine (159)–(163) to obtain the desired estimate:where is a constant depending on .
6. Numerical Example
Here, we give an illustration of our work by testing the algorithm on an example of the q(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 , showing that the error values decrease to zero in the norm when 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 and in the q(x)-Laplacian parabolic equation are as follows:
Consequently, the parabolic q(x)-Laplacian parabolic equation becomes the following couple of the linear Fokker–Planck equation and the -Laplacian parabolic equation:
One can assimilate the set to a nonhomogeneous composite rod of length 1 cast in two distinct materials and made of a main piece surrounded at its extremities by two other pieces and . The nonhomogeneity is characterized by the two different constant values taken by the variable exponent on the distinct pieces of the rod.
Let us express the solution of the linear Fokker–Planck equationin the formwhere is an unknown function, supposed to be a solution of the heat equation
Thus, after a simple computation, we have that
So, if satisfiesthen satisfies
Besides, the function is an exact solution of heat equation (169). Accordingly, the function defined byis an exact solution of equation (167).
Next, let us deal with the parabolic -Laplacian equation:
We obtain an exact solution of (174) in the form
Finally, fix the initial datum to be
Then, the exact solution 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 and as defined in the above, we deduce that the equationbecomes
Consequently, the function which is defined bysolves (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 for comparison with the exact solution, and we get
6.1. The Figures
For some values of , we have drawn the figures of both analytical and approximate solutions when step , using Scilab software. We mention that Figure 1 represents an approximation of Figure 2 when, Figure 3 represents an approximation of Figure 4 when , and Figure 5 represents an approximation of Figure 6 when .
6.2. Table of Error Progression
The Table of Error Progression is provide in the Table 1
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by IMSP under the CEA-SMA project grant.