Solvability of a Semilinear Parabolic Equation with Measures as Initial Data

Abstract

We study a sharp condition for the solvability of the Cauchy problem $u_t-\varDelta u=u^p$, $u(\cdot ,0)=\mu$, where $N\ge 1$, $p\ge (N+2)/N$ and $\mu$ is a Radon measure on $\mathbf {R}^N$. Our results show that the problem does not admit any local nonnegative solutions for some $\mu$ satisfying $\mu (\{y\in \mathbf {R}^N; |x-y|<\rho \}) \le C\rho ^{N-2/(p-1)}(\log (e+1/\rho ))^{-1/(p-1)}$ ($x \in \mathbf {R}^N$, $\rho >0$) with a constant $C>0$. On the other hand, the problem admits a local solution if $\mu (\{y\in \mathbf {R}^N; |x-y|<\rho \}) \le C\rho ^{N-2/(p-1)}(\log (e+1/\rho ))^{-1/(p-1)-\varepsilon }$ ($x \in \mathbf {R}^N$, $\rho >0$) with a constant $\varepsilon \in (0,1/(p-1))$.