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We show higher interior regularity for the Westervelt equation with strong nonlinear damping term ofq-Laplace type. Secondly, we investigate an interface coupling problem for these models, which arise, e.g. in the context of medical applications of high intensity focused ultrasound in the treatment of kidney stones. We show that the solution to the coupled problem exhibits piecewiseregularity in space, provided that the gradient of the acoustic pressure is essentially bounded in space and time on the whole domain. This result is of importance in numerical approximations of the present problem as well as in gradient based algorithms for finding the optimal shape of a focusing acoustic lens in lithotripsy. [ABSTRACT FROM PUBLISHER]

In this paper we consider the problem of identifying a coefficient function in the principal part of a nonlinear hyperbolic PDE from overdetermined boundary data of the PDE solution. Assuming that the coefficient function can be written as a linear combination of finitely many ansatz functions, we derive a stability and uniqueness result that is based on an identifiability condition in terms of the initial data as well as on the smallness of the time interval. In doing so, we distinguish between the two- and higher dimensional case where functionals on the boundary can be measured and the one-dimensional situation with only point measurements at one boundary point. Our results also give perspectives to the case of an infinite dimensional coefficient function. [ABSTRACT FROM AUTHOR]

In this paper, we propose a multilevel method for solving nonlinear ill-posed operator equations By minimizing the distance to some initial guess under the constraint of a discretized version of the operator equation for different levels of discretization, we define a sequence of regularized approximations to the exact solution, which is shown to be stable and convergent for arbitrary initial guess and can be computed via a multilevel procedure that altogether yields a globally convergent method. Moreover, this approach enables one to relax restrictions on the nonlinearity of the forward operator, as were used in previous work on regularization methods for nonlinear ill-posed problems. [ABSTRACT FROM AUTHOR]