Mixed boundary valued problems for linear and nonlinear wave equations in domains with fractal boundaries

Abstract

The weak well-posedness, with the mixed boundary conditions, of the strongly damped linear wave equation and of the non linear Westervelt equation is proved in a large natural class of Sobolev admissible non-smooth domains. In the framework of uniform domains in \(\mathbb {R}^2\) or \(\mathbb {R}^3\) we also validate the approximation of the solution of the Westervelt equation on a fractal domain by the solutions on the prefractals using the Mosco convergence of the corresponding variational forms.