Very Weak Solutions to the Boundary-Value Problem of the Homogeneous Heat Equation

Abstract

We consider the homogeneous heat equation in a domain $\Omega$ in ${\mathbb{R}}^n$ with vanishing initial data and Dirichlet boundary conditions. We are looking for solutions in $W_{p,q}^{r,s}(\Omega \times (0,T))$, where $r<2$, $s<1$, $1\le p<\infty$, $1\le q\le \infty$. Since we work in the $L_{p,q}$ framework any extension of the boundary data and integration by parts are not possible. Therefore, the solution is represented in integral form and is referred as \emph{very weak} solution. The key estimates are performed in the half-space and are restricted to $L_q(0,T;W_p^{\alpha }(\Omega ))$, $0\le \alpha <\frac{1}{p}$ and $L_q(0,T;W_p^{\alpha }(\Omega ))$, $\alpha \le 1$. Existence and estimates in the bounded domain $\Omega$ follow from a perturbation and a fixed point arguments.