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

Aboubacar Marcos,Ambroise Soglo

doi:10.1155/2020/9756162

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 ∂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.

Highlights

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: zρ(t, zt x) divx􏼐ρ(t, x)
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]
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 Wp(.) defined by zρ(t, x)

Summary

Introduction and the Main Results

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: zρ(t, zt x) divx􏼐ρ(t, x). (3) We use the maximum principle N ≤ ρk ≤ M and (10) to prove that the sequence (ρk)k is a time discretization of the nonlinear q(x)-Laplacian parabolic equation, that is, for all test functions, φ ∈ C∞ c (Rd),. We combine (i) and (ii) to prove that the sequence (ρh)h converges to a weak solution of the q(x)-Laplacian parabolic equation. We use the energy method to study the convergence of solutions of the q(x)-Laplacian parabolic equation to ρ∞ when t ⟶ ∞, where ρ∞ ∈ P(Ω) is the probability density which satisfies ρ∞∇x(G′(ρ∞) + V) 0.

Preliminaries

Asymptotic Behavior

Error Estimate

Numerical Example