Large time behavior of ODE type solutions to parabolic $ p $-Laplacian type equations

Junyong Eom; Ryuichi Sato

doi:10.3934/cpaa.2020199

Article Contents

2020, Volume 19, Issue 9: 4373-4386. Doi: 10.3934/cpaa.2020199

This issue Previous Article Monotonicity with respect to

$ p $

of the First Nontrivial Eigenvalue of the

$ p $

-Laplacian with Homogeneous Neumann Boundary Conditions Next Article Maximum principles for a fully nonlinear nonlocal equation on unbounded domains

Large time behavior of ODE type solutions to parabolic $ p $-Laplacian type equations

Junyong Eom^1, , and
Ryuichi Sato^2,

1.
Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan
2.
Mathematical Institute, Tohoku University, Aoba, Sendai 980-8578, Japan

^* Corresponding author
^* Corresponding author

Received: September 30, 2019

Revised: February 29, 2020

Published: June 2020

The second author was partially supported by Grant-in-Aid for Early-Career Scientists JSPS KAKENHI Grant Number 18K13435

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Abstract

Let $ u $ be a solution to the Cauchy problem for a nonlinear diffusion equation

$ \begin{equation*} \begin{cases} \partial_t u = \mathrm{div}\, (|\nabla u|^{p-2} \nabla u) + u^\alpha & \quad\mathrm{in}\quad{\bf R}^N\times(0, \infty), \\ u(x, 0) = \lambda+\varphi(x) & \quad\mathrm{in}\quad{\bf R}^N, \end{cases} \end{equation*} $

where $ N \ge 1 $, $ 2N/(N+1)<p\neq2 $, $ \alpha \in (-\infty, 1) $, $ \lambda>0 $ and $ \varphi\in BC({\bf R}^N)\, \cap\, L^1({\bf R}^N) $ with $ \varphi\geq0 $ in $ {\bf R}^{N} $. Then the solution $ u $ behaves like a positive solution to ODE $ \zeta' = \zeta^\alpha $ in $ (0, \infty) $. In this paper we show that the large time behavior of the solution $ u $ is described by a rescaled Barenblatt solution.

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Mathematics Subject Classification: Primary: 35B40, 35K55, 35K92.

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References

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