Existence for singular doubly nonlinear systems of porous medium type with time dependent boundary values

摘要

In this paper we prove the existence of variational solutions to the Cauchy–Dirichlet problem with time dependent boundary values associated with doubly nonlinear systems ?t(|u|m-1u)-div(Dξf(Du))=0documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$begin{aligned} partial _t big (|u|^{m-1}ubig ) - {{,mathrm{div},}}(D_xi f(Du)) = 0 end{aligned}$$end{document}with m>1documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$m>1$$end{document} and a convex function f satisfying a standard p-growth condition for an exponent p∈(1,∞)documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$p in (1,infty )$$end{document}. The proof relies on a nonlinear version of the method of minimizing movements.