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

• Published in: arXiv.org 2024-07
• Main Authors: Hisa, Kotaro, Kojima, Mizuki
• Format: Article
• Language: English
• Subjects: Thermodynamics
• ISSN: 2331-8422

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 \(1-0\).