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

  • Bernard Nowakowski

    Polish Academy of Sciences, Warszawa, Poland
  • Woiciech M. Zajączkowski

    Polish Academy of Sciences, Warszawa, Poland

Abstract

We consider the homogeneous heat equation in a domain in with vanishing initial data and Dirichlet boundary conditions. We are looking for solutions in , where , , , . Since we work in the 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 , and , . Existence and estimates in the bounded domain follow from a perturbation and a fixed point arguments.

Cite this article

Bernard Nowakowski, Woiciech M. Zajączkowski, Very Weak Solutions to the Boundary-Value Problem of the Homogeneous Heat Equation. Z. Anal. Anwend. 32 (2013), no. 2, pp. 129–153

DOI 10.4171/ZAA/1477