Existence of weak solutions to doubly degenerate diffusion equations
Aleš Matas, Jochen Merker · Applications of Mathematics · 2012
We prove existence of weak solutions to doubly degenerate diffusion equations $\dot u = \Delta _p u^{m - 1} + f(m,p \geqslant 2)$ 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 Ω ⊂ 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.