Existence of solutions for a fractional semilinear parabolic equation with singular initial data

By Kotaro Hisa, Kazuhiro Ishige. Nonlinear Analysis, 2016

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In this paper we obtain necessary conditions and sufficient conditions on the initial data for the solvability of the Cauchy problem $$ \partial_t u+(-\Delta)^{\frac{\theta}{2}}u=u^p,\quad x\in{\bf R}^N,\,\,t>0, \qquad u(0)=\mu\ge 0\quad\mbox{in}\quad{\bf R}^N, $$ where $N\ge 1$, $0 1$ and $\mu$ is a Radon measure or a measurable function in ${\bf R}^N$. Our conditions lead optimal estimates of the life span of the solution with $\mu$ behaving like $\lambda |x|^{-A}$ ($A>0$) at the space infinity, as $\lambda \to +0$.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/ba5928514f04a8f13977adea63263090f8ee6688