Existence of weak solutions to doubly degenerate diffusion equations

Aleš Matas; Jochen Merker

Applications of Mathematics (2012)

  • Volume: 57, Issue: 1, page 43-69
  • ISSN: 0862-7940

Abstract

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We prove existence of weak solutions to doubly degenerate diffusion equations u ˙ = Δ p u m - 1 + f ( m , p ≥ 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.

How to cite

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Matas, Aleš, and Merker, Jochen. "Existence of weak solutions to doubly degenerate diffusion equations." Applications of Mathematics 57.1 (2012): 43-69. <http://eudml.org/doc/246610>.

@article{Matas2012,
abstract = {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 $\operatorname\{div\}(F(u))$. In the appendix, an introduction to weak differentiability of functions with values in a Banach space appropriate for doubly nonlinear evolution equations is given.},
author = {Matas, Aleš, Merker, Jochen},
journal = {Applications of Mathematics},
keywords = {$p$-Laplacian; doubly nonlinear evolution equation; weak solution; -Laplacian; doubly nonlinear evolution equation; weak solution},
language = {eng},
number = {1},
pages = {43-69},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Existence of weak solutions to doubly degenerate diffusion equations},
url = {http://eudml.org/doc/246610},
volume = {57},
year = {2012},
}

TY - JOUR
AU - Matas, Aleš
AU - Merker, Jochen
TI - Existence of weak solutions to doubly degenerate diffusion equations
JO - Applications of Mathematics
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 57
IS - 1
SP - 43
EP - 69
AB - 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 $\operatorname{div}(F(u))$. In the appendix, an introduction to weak differentiability of functions with values in a Banach space appropriate for doubly nonlinear evolution equations is given.
LA - eng
KW - $p$-Laplacian; doubly nonlinear evolution equation; weak solution; -Laplacian; doubly nonlinear evolution equation; weak solution
UR - http://eudml.org/doc/246610
ER -

References

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