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

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.