Article
MSC:
35A01,
35D30,
35K20,
35K59,
35K65,
35K92,
37L65 | MR 2891305 | Zbl 1249.35194 | DOI: 10.1007/s10492-012-0004-0
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Keywords:
$p$-Laplacian; doubly nonlinear evolution equation; weak solution
$p$-Laplacian; doubly nonlinear evolution equation; weak solution
Summary:
We prove existence of weak solutions to doubly degenerate diffusion equations \begin {equation*} \dot {u} = \Delta _p u^{m-1} + f \quad (m,p \ge 2) \end {equation*} 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 $\div (F(u))$. In the appendix, an introduction to weak differentiability of functions with values in a Banach space appropriate for doubly nonlinear evolution equations is given.
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