On solvability of a time-fractional semilinear heat equation, and its quantitative approach to the classical counterpart

TL;DR

This paper investigates the solvability of a time-fractional semilinear heat equation, especially at the critical Fujita exponent, and explores how solutions transition as the fractional order approaches the classical case.

Contribution

It provides a detailed analysis of the solvability transition of the fractional heat equation at the Fujita critical exponent as the fractional order approaches 1.

Findings

Time-fractional equation is globally solvable at critical exponent, unlike the classical case.

The solvability properties change continuously as the fractional order approaches 1.

The paper clarifies the collapse of global and local solvability in the limit.

Abstract

We are concerned with the following time-fractional semilinear heat equation in the $N$-dimensional whole space ${\bf R}^N$ with $N \geq 1$. \[ {\rm (P)}_\alpha \qquad \partial_t^\alpha u -\Delta u = u^p,\quad t>0,\,\,\, x\in{\bf R}^N, \qquad u(0) = \mu \quad \mbox{in}\quad {\bf R}^N, \] where $\partial_t^\alpha$ denotes the Caputo derivative of order $\alpha \in (0,1)$, $p>1$, and $\mu$ is a nonnegative Radon measure on ${\bf R}^N$. The case $\alpha=1$ formally gives the Fujita-type equation (P)$_1$ \ $\partial_tu-\Delta u=u^p$. In particular, we mainly focus on the Fujita critical case where $p=p_F:=1+2/N$. It is well known that the Fujita exponent $p_F$ separates the ranges of $p$ for the global-in-time solvability of (P)$_1$. In particular, (P)$_1$ with $p=p_F$ possesses no global-in-time solutions, and does not locally-in-time solvable in its scale critical space $L^1(\mathbf{R}^N)$. It is also known that the exponent $p_F$ plays the same role for the global-in-time solvability for (P)$_\alpha$. However, the problem (P)$_\alpha$ with $p=p_F$ is globally-in-time solvable, and exhibites local-in-time solvability in its scale critical space $L^1(\mathbf{R}^N)$. The purpose of this paper is to clarify the collapse of the global and local-in-time solvability of (P)$_\alpha$ as $\alpha$ approaches $1-0$.