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# Zeitschrift für Analysis und ihre Anwendungen

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**Volume 32, Issue 2, 2013, pp. 129–153**

**DOI: 10.4171/ZAA/1477**

Published online: 2013-04-08

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

Bernard Nowakowski^{[1]}and Woiciech M. Zajączkowski

^{[2]}(1) Polish Academy of Sciences, Warszawa, Poland

(2) Polish Academy of Sciences, Warszawa, Poland

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 $W^{r,s}_{p,q}(\Omega\times(0,T))$, where $r < 2$, $s < 1$, $1 \leq p < \infty$, $1 \leq q \leq \infty$. 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 $L_q(0,T;W^{\alpha}_p(\Omega))$, $0 \leq \alpha < \frac{1}{p}$ and $L_q(0,T;W^{\alpha}_p(\Omega))$, $\alpha \leq 1$. Existence and estimates in the bounded domain $\Omega$ follow from a perturbation and a fixed point arguments.

*Keywords: *Very weak solutions, integral equation, non-regular solutions, anisotropic Sobolev-Slobodecki

Nowakowski Bernard, Zajączkowski Woiciech: Very Weak Solutions to the Boundary-Value Problem of the Homogeneous Heat Equation. *Z. Anal. Anwend.* 32 (2013), 129-153. doi: 10.4171/ZAA/1477