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

Abstract: The weak well-posedness results of the strongly damped linear wave equation and of the non linear Westervelt equation with homogeneous Dirichlet boundary conditions are proved on arbitrary three dimensional domains or any two dimensional domains which can be obtained by a limit of NTA domains caractarized by the same geometrical constants. The two dimensional result is obtained thanks to the Mosco convergence of the functionals corresponding to the weak formulations for the Westervelt equation with the homogeneous Dirichlet boundary condition. The non homogeneous Dirichlet condition is also treated in the class of admissible domains composed on Sobolev extension domains of $\mathbb{R}^n$ with a $d$-set boundary $n-1\leq d<n$ preserving Markov's local inequality.The obtained Mosco convergence also alows to approximate the solution of the Westervelt equation on an arbitrary domain by solutions on a converging sequence of domains without additional conditions on their boundary regularity in $\mathbb{R}^3$, or on a converging sequence of NTA domains in $\mathbb{R}^2$.