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

2020 ◽  
Vol 2020 ◽  
pp. 1-23
Author(s):  
Aboubacar Marcos ◽  
Ambroise Soglo
Keyword(s):  
Asymptotic Behavior ◽  
Parabolic Equation ◽  
Parabolic Equations ◽  
Variable Exponent ◽  
Probability Space ◽  
Compact Embedding ◽  
Convergence Of Solutions ◽  
Existence Of Positive Solutions ◽  
Wasserstein Space ◽  
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 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.

scholarly journals Non-autonomous perturbations of a non-classical non-autonomous parabolic equation with subcritical nonlinearity

2017 ◽  
Vol 2 (1) ◽  
pp. 31-60 ◽  
Author(s):  
Matheus C. Bortolan ◽  
Felipe Rivero
Keyword(s):  
Asymptotic Behavior ◽  
Parabolic Equation ◽  
Bounded Domain ◽  
Small Perturbation ◽  
Parabolic Equations ◽  
Smooth Bounded Domain ◽  
Subcritical Nonlinearity

AbstractIn this work we study the continuity of four different notions of asymptotic behavior for a family of non-autonomous non-classical parabolic equations given by$$\begin{array}{} \displaystyle \left\{ \begin{array}{*{20}{l}} {{u_t} - \gamma \left( t \right)\Delta {u_t} - \Delta u = {g_\varepsilon }\left( {t,u} \right),{\;\text{in}\;}\Omega } \hfill \\ {u = 0,{\;\text{on}\;}\partial \Omega {\rm{.}}} \hfill \\ \end{array}\right. \end{array}$$in a smooth bounded domain Ω ⊂ ℝn, n ⩾ 3, where the terms gε are a small perturbation, in some sense, of a function f that depends only on u.

scholarly journals Hölder gradient estimates for a class of singular or degenerate parabolic equations

2017 ◽  
Vol 8 (1) ◽  
pp. 845-867 ◽  
Author(s):  
Cyril Imbert ◽  
Tianling Jin ◽  
Luis Silvestre
Keyword(s):  
Parabolic Equation ◽  
Parabolic Equations ◽  
Degenerate Parabolic Equation ◽  
Divergence Form ◽  
Gradient Estimates ◽  
Degenerate Parabolic Equations ◽  
Spatial Gradients ◽  
Degenerate Parabolic ◽  
Laplacian Equations ◽  
Hölder Estimates

Abstract We prove interior Hölder estimates for the spatial gradients of the viscosity solutions to the singular or degenerate parabolic equation u_{t}=\lvert\nabla u\rvert^{\kappa}\operatorname{div}(\lvert\nabla u\rvert^{p-% 2}\nabla u), where {p\in(1,\infty)} and {\kappa\in(1-p,\infty)} . This includes the from {L^{\infty}} to {C^{1,\alpha}} regularity for parabolic p-Laplacian equations in both divergence form with {\kappa=0} , and non-divergence form with {\kappa=2-p} .

scholarly journals Weak Solutions for Nonlinear Parabolic Equations with Variable Exponents

2017 ◽  
Vol 25 (1) ◽  
pp. 55-70 ◽  
Author(s):  
Lingeshwaran Shangerganesh ◽  
Arumugam Gurusamy ◽  
Krishnan Balachandran
Keyword(s):  
Parabolic Equation ◽  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
Variable Exponent ◽  
Degenerate Parabolic Equation ◽  
Nonlinear Parabolic Equations ◽  
Variable Exponents ◽  
Nonlinear Parabolic ◽  
Order Degenerate ◽  
Degenerate Parabolic

Abstract In this work, we study the existence and uniqueness of weak solu- tions of fourth-order degenerate parabolic equation with variable exponent using the di erence and variation methods.

Hypothesis on the Solvability of Parabolic Equations with nonlocal Condition

2002 ◽  
Vol 7 (1) ◽  
pp. 93-104 ◽  
Author(s):  
Mifodijus Sapagovas
Keyword(s):  
Parabolic Equation ◽  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
Nonlocal Condition ◽  
Nonlocal Conditions ◽  
Numerical Algorithms ◽  
Uniqueness Of The Solution

Numerous and different nonlocal conditions for the solvability of parabolic equations were researched in many articles and reports. The article presented analyzes such conditions imposed, and observes that the existence and uniqueness of the solution of parabolic equation is related mainly to ”smallness” of functions, involved in nonlocal conditions. As a consequence the hypothesis has been made, stating the assumptions on functions in nonlocal conditions are related to numerical algorithms of solving parabolic equations, and not to the parabolic equation itself.

Existence and asymptotical behavior of solutions for a quasilinear Choquard equation with singularity

2021 ◽  
Vol 19 (1) ◽  
pp. 259-267
Author(s):  
Liuyang Shao ◽  
Yingmin Wang
Keyword(s):  
Asymptotic Behavior ◽  
Positive Solutions ◽  
Riesz Potential ◽  
Asymptotic Behavior Of Solutions ◽  
Suitable Assumption ◽  
Choquard Equation ◽  
Existence Of Positive Solutions

Abstract In this study, we consider the following quasilinear Choquard equation with singularity − Δ u + V ( x ) u − u Δ u 2 + λ ( I α ∗ ∣ u ∣ p ) ∣ u ∣ p − 2 u = K ( x ) u − γ , x ∈ R N , u > 0 , x ∈ R N , \left\{\begin{array}{ll}-\Delta u+V\left(x)u-u\Delta {u}^{2}+\lambda \left({I}_{\alpha }\ast | u{| }^{p})| u{| }^{p-2}u=K\left(x){u}^{-\gamma },\hspace{1.0em}& x\in {{\mathbb{R}}}^{N},\\ u\gt 0,\hspace{1.0em}& x\in {{\mathbb{R}}}^{N},\end{array}\right. where I α {I}_{\alpha } is a Riesz potential, 0 < α < N 0\lt \alpha \lt N , and N + α N < p < N + α N − 2 \displaystyle \frac{N+\alpha }{N}\lt p\lt \displaystyle \frac{N+\alpha }{N-2} , with λ > 0 \lambda \gt 0 . Under suitable assumption on V V and K K , we research the existence of positive solutions of the equations. Furthermore, we obtain the asymptotic behavior of solutions as λ → 0 \lambda \to 0 .

Intrinsic scaling method for doubly nonlinear parabolic equations and its application

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Masashi Misawa ◽  
Kenta Nakamura
Keyword(s):  
Parabolic Equation ◽  
Parabolic Equations ◽  
Large Data ◽  
Nonlinear Parabolic Equations ◽  
Sobolev Type ◽  
Scaling Method ◽  
Nonlinear Parabolic ◽  
Doubly Nonlinear ◽  
Doubly Nonlinear Equation ◽  
Doubly Nonlinear Parabolic Equation

Abstract In this article, we consider a fast diffusive type doubly nonlinear parabolic equation, called 𝑝-Sobolev type flows, and devise a new intrinsic scaling method to transform the prototype doubly nonlinear equation to the 𝑝-Sobolev type flows. As an application, we show the global existence and regularity for the 𝑝-Sobolev type flows with large data.

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