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Existence of weak solutions to doubly degenerate diffusion equations. (English) Zbl 1249.35194
Summary: s: We prove existence of weak solutions to doubly degenerate diffusion equations $$\dot {u}=\Delta _pu^{m-1}+f$$ ($$m,p\geq 2$$) by Faedo-Galerkin approximation for general domains and general nonlinearities. More precisely, we discuss the equation in an abstract setting, which allows to choose function spaces corresponding to bounded or unbounded domains $$\Omega \subset \mathbb {R}^n$$ with Dirichlet or Neumann boundary conditions. The function $$f$$ can be an inhomogeneity or a nonlinearity involving terms of the form $$f(u)$$ or $$\text{div}(F(u))$$. In the appendix, an introduction to weak differentiability of functions with values in a Banach space appropriate for doubly nonlinear evolution equations is given.

##### MSC:
 35K92 Quasilinear parabolic equations with $$p$$-Laplacian 35D30 Weak solutions to PDEs 37L65 Special approximation methods (nonlinear Galerkin, etc.) for infinite-dimensional dissipative dynamical systems
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