Brezis-Merle type inequality for a heat equation in two dimensions

Abstract

We consider the regularity estimate for the solution of the inhomogeneous heat equation in $(0,T)\times \Omega$ with 0-Dirichlet boundary conditions, where $ \Omega$ is a bounded domain in $\mathbb{R}^2$. We choose the external force $f$ from the Lorentz-Zygmund space which is not a reflexive Banach space, therefore our estimate cannot be obtained by a simple application of the estimate of the maximal regularity.