JournalszaaVol. 32, No. 2pp. 129–153

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

We consider the homogeneous heat equation in a domain Ω\Omega in Rn\mathbb{R}^n with vanishing initial data and Dirichlet boundary conditions. We are looking for solutions in Wp,qr,s(Ω×(0,T))W^{r,s}_{p,q}(\Omega\times(0,T)), where r<2r < 2, s<1s < 1, 1p<1 \leq p < \infty, 1q1 \leq q \leq \infty. Since we work in the Lp,qL_{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 Lq(0,T;Wpα(Ω))L_q(0,T;W^{\alpha}_p(\Omega)), 0α<1p0 \leq \alpha < \frac{1}{p} and Lq(0,T;Wpα(Ω))L_q(0,T;W^{\alpha}_p(\Omega)), α1\alpha \leq 1. Existence and estimates in the bounded domain Ω\Omega 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