Doubly nonlinear parabolic equation with perturbation

TL;DR

This paper studies a doubly nonlinear parabolic equation with perturbation, aiming to relax growth conditions and apply results to singular and degenerate cases, using energy estimates and discussing solution uniqueness.

Contribution

It introduces a new approach to handle nonlinear perturbations in doubly nonlinear parabolic equations with relaxed growth conditions.

Findings

Established $L^{inity}$-estimates for time-discrete solutions

Extended applicability to singular and degenerate equations

Proved uniqueness of solutions under new conditions

Abstract

In this paper, we consider the initial boundary value problem of a doubly nonlinear parabolic equation with nonlinear perturbation. We impose the homogeneous Dirichlet condition on this problem. We aim to reduce the growth condition of the nonlinear term and the largeness of exponent as possible so that we can apply our result to both singular and degenerate type parabolic equations. We use in our proof the $L ^{\infty }$-estimate of the time-discrete equation derived in the previous work and apply the so-called $L ^{\infty }$-energy method to the time-discrete problem. We also discuss the uniqueness of solution by using the previous result.