The Wave Equation in Lp-Spaces

Abstract

In the first part of the paper, Gaussian estimates are used to study $L^p$-summability of the solution of the wave equation in $L^p(\Omega)$ associated with a general operator in divergence form with bounded coefficients. Secondly, we prove that if $\Omega$ is a cube in $\RR^N$, then the Laplacian with Dirichlet or Neumann boundary conditions generates an $\al$-times integrated cosine function on $L^p(\Omega),\;1\le p <\infty$ for any $\al\ge (N-1)|\frac{1}{2}-\frac{1}{p}|$.