Solvability of inhomogeneous fractional semilinear heat equations in Lorentz--Morrey spaces

TL;DR

This work addresses the local-in-time solvability of the fractional semilinear heat equation $\partial_t u+(-\Delta)^{\theta/2}u=|u|^{\gamma-1}u+\mu$ with distributional inhomogeneity $\mu$, by introducing Lorentz–Morrey spaces and Besov–Lorentz–Morrey (BLM) spaces to capture sign-changing sources and singular data. The authors establish sharp integral estimates for the nonlinear term and prove local solvability under three Besov-type conditions on $\mu$, along with a necessary condition that $\mu$ lie in $B^{-{\theta},\infty}_{p,q,\infty}$ if a solution exists in $L^{\infty}(0,T;M^{p}_{q,\infty})$; a key innovation is a Besov-type norm estimate for inhomogeneous terms via the time-integrated heat semigroup. The analysis combines decay estimates for $S(t)$ in the Lorentz–Morrey and BLM spaces, real interpolation tailored to Morrey-type settings, and a contraction-mapping framework to treat both the Serrin-supercritical and -subcritical regimes. The results extend solvability to distributional inhomogeneities such as derivatives of the Dirac delta, enabling broader applicability to singular sources in fractional parabolic problems.

Abstract

We study the Cauchy problem for the fractional semilinear heat equation with distributional inhomogeneous terms. By introducing the Lorentz--Morrey spaces, we overcome limitations of real interpolation in the classical local Morrey spaces and obtain a sharp integral estimate for the nonlinear term. Moreover, in terms of Besov-type spaces, we give necessary conditions and sufficient conditions on inhomogeneous terms for the local-in-time existence of solutions belonging to Lorentz--Morrey spaces.

Theorems & Definitions (59)

• Definition 1.1
• Remark 1.2
• Definition 1.3
• Definition 1.4
• Theorem 1.5
• Remark 1.6
• Remark 1.7
• Remark 1.8
• Remark 1.9
• Theorem 1.10