Large time behavior of ODE type solutions to nonlinear diffusion equations
Large time behavior of ODE type solutions to nonlinear diffusion equations
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DOI：
10.3934/dcds.2019229
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发表时间：
2018-09
期刊：
影响因子：
--
通讯作者：
Junyong Eom;Kazuhiro Ishige
中科院分区：
文献类型：
--
作者：
Junyong Eom;Kazuhiro Ishige
Consider the Cauchy problem for a nonlinear diffusion equation P \begin{document}$ \begin{equation} \left\{ \begin{array}{ll} \partial_t u = \Delta u^m+u^\alpha & \quad\mbox{in}\quad{\bf R}^N\times(0,\infty),\\ u(x,0) = \lambda+\varphi(x)>0 & \quad\mbox{in}\quad{\bf R}^N, \end{array} \right. \end{equation} $\end{document} where \begin{document}$ m>0 $\end{document} , \begin{document}$ \alpha\in(-\infty,1) $\end{document} , \begin{document}$ \lambda>0 $\end{document} and \begin{document}$ \varphi\in BC({\bf R}^N)\,\cap\, L^r({\bf R}^N) $\end{document} with \begin{document}$ 1\le r and \begin{document}$ \inf_{x\in{\bf R}^N}\varphi(x)>-\lambda $\end{document} . Then the positive solution to problem (P) behaves like a positive solution to ODE \begin{document}$ \zeta' = \zeta^\alpha $\end{document} in \begin{document}$ (0,\infty) $\end{document} and it tends to \begin{document}$ +\infty $\end{document} as \begin{document}$ t\to\infty $\end{document} . In this paper we obtain the precise description of the large time behavior of the solution and reveal the relationship between the behavior of the solution and the diffusion effect the nonlinear diffusion equation has.
Consider the Cauchy problem for a nonlinear diffusion equation P \begin{document}$ \begin{equation} \left\{ \begin{array}{ll} \partial_t u = \Delta u^m+u^\alpha & \quad\mbox{in}\quad{\bf R}^N\times(0,\infty),\\ u(x,0) = \lambda+\varphi(x)>0 & \quad\mbox{in}\quad{\bf R}^N, \end{array} \right. \end{equation} $\end{document} where \begin{document}$ m>0 $\end{document} , \begin{document}$ \alpha\in(-\infty,1) $\end{document} , \begin{document}$ \lambda>0 $\end{document} and \begin{document}$ \varphi\in BC({\bf R}^N)\,\cap\, L^r({\bf R}^N) $\end{document} with \begin{document}$ 1\le r and \begin{document}$ \inf_{x\in{\bf R}^N}\varphi(x)>-\lambda $\end{document} . Then the positive solution to problem (P) behaves like a positive solution to ODE \begin{document}$ \zeta' = \zeta^\alpha $\end{document} in \begin{document}$ (0,\infty) $\end{document} and it tends to \begin{document}$ +\infty $\end{document} as \begin{document}$ t\to\infty $\end{document} . In this paper we obtain the precise description of the large time behavior of the solution and reveal the relationship between the behavior of the solution and the diffusion effect the nonlinear diffusion equation has.