Local Well-Posedness for the Cauchy Problem to Nonlinear Heat Equations of Fujita Type in Nearly Critical Besov Space

Abstract

We show the local well-posedness of the Cauchy problem to a nonlinear heat equation of Fujita type in lower space dimensions. It is well known that the nonnegative solution corresponding to the Fujita critical exponent \(p=1+\frac{2}{n}\) does not exist in the critical scaling invariant space \(L^1(\mathbb R^n)\). We show if the initial data is in a modified Besov spaces, then the corresponding mild solution to the equation with the Fujita critical exponent \(p=1+\frac{2}{n}\) exists and the problem is locally well-posed in the same space of the initial data. Besides we also show the problem is ill-posed in the scaling invariant Besov and inhomogeneous Besov spaces. This is known in \(L^1\) space and extension of the result known in the Lebesgue spaces.