Large time behavior of ODE type solutions to parabolic $ p $ -Laplacian type equations
Junyong Eom, Ryuichi Sato · Communications on Pure & Applied Analysis · 2020
Let $ u $ be a solution to the Cauchy problem for a nonlinear diffusion equation \begin{document}$ \begin{equation*} \begin{cases} \partial_t u = \mathrm{div}\, (| abla u|^{p-2} abla 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*} $\end{document} where $ N \ge 1 $, $ 2N/(N+1) 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.