Existence of a Sign-Changing Weak Solution to Doubly Nonlinear Parabolic Equations

Abstract

In this paper, assuming the initial-boundary datum belonging to suitable Sobolev and Lebesgue spaces, we prove the global existence result for a (possibly sign changing) weak solution to the Cauchy–Dirichlet problem for doubly nonlinear parabolic equations of the form ∂t|u|q-1u-Δpu=0inΩ∞,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\partial _t\\left( |u|^{q-1}u\\right) -\\Delta _p u=0\\quad \ ext {in}\\,\\,\\,\\Omega _\\infty , \\end{aligned}$$\\end{document}where p>1 and q>0. This is a fair improvement of the preceding result by authors (Nonlinear Anal 175C :157–172, 2018). The key tools we employ are energy estimates for approximate equations of Rothe type and the integral strong convergence of gradients of approximate solutions.