Description

• Title:
  The blow-up and global existence of solutions of Cauchy problems for a time fractional diffusion equation
• Creator:
  Zhang, Quan-Guo;
  Sun, Hong-Rui
• Description:
  In this paper, we investigate the blow-up and global existence of solutions to the following time fractional nonlinear diffusion equations {_0^C D_t^\alpha u}-\triangle u=|u|^{p-1}u, x\in \Bbb{R}^N,\ t\ge0, u(0,x)=u_0(x), x\in \Bbb{R}^N, where $0\le\alpha\le 1$, $p\ge 1$, $u_0\in C_0(\Bbb{R}^N)$ and ${_0^CD_t^\alpha u}=({\partial}/{\partial t}){_0^{}I_t^{1-\alpha}(u(t,x)-u_0(x))}$, ${_0^{}I_t^{1-\alpha}}$ denotes left Riemann--Liouville fractional integrals of order $1-\alpha$. We prove that if $1\le p\le 1+{2}/{N}$, then every nontrivial nonnegative solution blow-up in finite time, and if $p\geq 1+{2}/{N}$ and $\|u_0\|_{L^{q_c}(\Bbb{R}^N)}$, $q_c={N(p-1)}/{2}$ is sufficiently small, then the problem has global solution.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2015 Topological Methods in Nonlinear Analysis
• Date:
  2015.09.01
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 46, No 1 (September 2015); 69 - 92;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2015.038/7201
• Subject:
  Fractional differential equation;
  blow-up;
  global existence;
  Cauchy problems;
  26A33;
  35K15
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2015.038
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other