On solvability of a time-fractional doubly critical semilinear equation, and its quantitative approach to the non-existence result on the classical counterpart

TL;DR

This paper investigates a time-fractional semilinear heat equation at the critical exponent, revealing that fractional derivatives enable solutions where classical equations do not, and provides conditions on initial data for solvability.

Contribution

It establishes necessary conditions for solutions and explores how fractional derivatives affect solvability compared to classical equations.

Findings

Time-fractional equation admits solutions where classical counterparts do not.

Necessary conditions on initial data for existence of solutions.

Solvability collapses as fractional order approaches 1.

Abstract

We study a time-fractional semilinear heat equation $$\partial^{\alpha}_t u -\Delta u = u^{p},\ \ \mbox{in}\ (0,T)\times\mathbb{R}^N,\ \ u(0)=u_0\ge0$$ with $u_0\in L^{1}(\mathbb{R}^N)$ and $p=1+2/N$. Here $\partial_t^{\alpha}$ denotes the Caputo derivative of order $\alpha \in (0,1)$. Since the space $L^1(\mathbb{R}^N)$ is scale critical with $p=1+2/N$, this type of equation is known as a doubly critical problem. It is known that the usual doubly critical equation $\partial_t u-\Delta u=u^p$ does not have nonnegative global-in-time solutions, while the time-fractional problem does. Moreover, there exists a singular initial data which admits no local-in-time solution, while the time-fractional equation is solvable for any $L^{1}(\mathbb{R}^N)$ initial data. In this paper, we deduce a necessary condition imposed on $u_0$ for the existence of a nonnegative solution. Furthermore, we obtain corollaries that describe the collapse of the local and global solvability for the time-fractional equation as $\alpha \rightarrow 1$.