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

Abstract

AbstractIn 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 $$\begin{aligned} \partial _t\left( |u|^{q-1}u\right) -\Delta _p u=0\quad \text {in}\,\,\,\Omega _\infty , \end{aligned}$$ ∂ t | u | q - 1 u - Δ p u = 0 in Ω ∞ , where $$p>1$$ p > 1 and $$q>0$$ 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.