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We consider the homogenization of nonlinear random parabolic operators. Depending on the ratio between time and spatial scales different homogenization regimes are studied and the homogenization procedure is carried out. The parameter dependent auxiliary problem is investigated and used in the construction of the homogenized operator.
In this paper we prove local existence of weak solutions for a semilinear wave equation with power-like source and dissipative terms on the entire space $\mathbb R^n$. The main theorem gives an alternative proof of the local in time existence result due to J. Serrin, G. Todorova and E. Vitillaro, and also some extension to their work. In particular, our method shows that sources that are not locally Lipschitz in $L^2$ can be controlled without any damping at all. If the semilinearity involving the displacement has a "good" sign, we obtain global existence of solutions.
In this note we introduce a general framework which allows us to prove in a unified and systematic way that operators with Wentzell-type boundary conditions generate analytic semigroups on function spaces with bounded trace operator. The abstract generation result is illustrated in three concrete examples.