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 ∂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 probabi...
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| Format: | Article |
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Wiley
2020-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2020/9756162 |
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| author | Aboubacar Marcos Ambroise Soglo |
| author_facet | Aboubacar Marcos Ambroise Soglo |
| author_sort | Aboubacar Marcos |
| collection | DOAJ |
| description | In this paper, we extend the variational method of M. Agueh to a large class of parabolic equations involving q(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. |
| format | Article |
| id | doaj-art-a8caa698868443f6951a353db4590b58 |
| institution | Kabale University |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2020-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-a8caa698868443f6951a353db4590b582025-08-20T03:39:09ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2020-01-01202010.1155/2020/97561629756162Existence of Positive Solutions and Asymptotic Behavior for Evolutionary q(x)-Laplacian EquationsAboubacar Marcos0Ambroise Soglo1Institut de Mathématiques et de Sciences Physiques, Université d’Abomey-Calavi, Cotonou, BeninInstitut de Mathématiques et de Sciences Physiques, Université d’Abomey-Calavi, Cotonou, BeninIn this paper, we extend the variational method of M. Agueh to a large class of parabolic equations involving q(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.http://dx.doi.org/10.1155/2020/9756162 |
| spellingShingle | Aboubacar Marcos Ambroise Soglo Existence of Positive Solutions and Asymptotic Behavior for Evolutionary q(x)-Laplacian Equations Discrete Dynamics in Nature and Society |
| title | Existence of Positive Solutions and Asymptotic Behavior for Evolutionary q(x)-Laplacian Equations |
| title_full | Existence of Positive Solutions and Asymptotic Behavior for Evolutionary q(x)-Laplacian Equations |
| title_fullStr | Existence of Positive Solutions and Asymptotic Behavior for Evolutionary q(x)-Laplacian Equations |
| title_full_unstemmed | Existence of Positive Solutions and Asymptotic Behavior for Evolutionary q(x)-Laplacian Equations |
| title_short | Existence of Positive Solutions and Asymptotic Behavior for Evolutionary q(x)-Laplacian Equations |
| title_sort | existence of positive solutions and asymptotic behavior for evolutionary q x laplacian equations |
| url | http://dx.doi.org/10.1155/2020/9756162 |
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