Dirichlet boundary valued problems for linear and nonlinear wave equations on arbitrary and fractal domains

Abstract

We obtain the weak well-posedness results for the linear strongly damped wave equation and the nonlinear Westervelt equation on arbitrary three-dimensional domains with homogeneous Dirichlet boundary conditions. In R2, we prove the well-posedness in the class of NTA domains or their limit domains, obtained as a limit of sequences of NTA domains, characterized by the same geometrical constants. The nonhomogeneous Dirichlet condition is also treated for Sobolev extension domains of Rn with a d-set boundary n−2<d<n preserving Markov's local inequality. For a converging sequence of domains in the sense of characteristic functions, we establish the Mosco convergence of the functionals corresponding to the weak formulations for the Westervelt equation with the homogeneous Dirichlet boundary condition.