| Abstract: |
| We obtain the global on-time well-posedness of the Robin type boundary valued problem for the Westervelt equation on a bounded domain with a non-Lipschitz boundary. The obtained weak solutions are considered in the domain of Laplacian and thus are more regular than $H^1$. The irregularity of the boundary does not allow the usual $H^2$-regularity. 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. We consider the shape optimization problem for this ultrasound wave propagation model to minimize the system`s total acoustic energy by the shape of the boundary for fixed source and initial data. For the Robin boundary, modeling the reflection, we prove the existence of an optimal shape realizing the infimum of the acoustic energy in a class of Lipschitz boundaries. Using its relaxation on the uniform class of domains, we prove the exiistence of an optimal shape realizing the minimum. |
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