Renormalized Solutions for a Class of Nonlinear Parabolic Equation with a Lower Order Term and Variable Exponents

TL;DR

This paper establishes the existence of renormalized solutions for a class of nonlinear parabolic equations with variable exponents and a lower order term, using truncation methods, monotone operator theory, and gradient estimates.

Contribution

It introduces a novel approach to prove existence of solutions without coercivity conditions in variable exponent frameworks.

Findings

Proves existence of renormalized solutions for complex nonlinear parabolic equations.

Develops new techniques combining truncation, monotone operators, and gradient estimates.

Handles lower order terms with variable growth conditions.

Abstract

We consider a class of nonlinear parabolic equations \[ \dfrac{\partial}{\partial t} b(u)-\nabla \cdot (A(x,t,u,\nabla u))+H(x,t,\nabla u)=f , \] where $H$ is a nonlinear lower order term satisfied the Carath$\acute{e}$odory condition and \[ \left\lvert H(x,t,\nabla u)\right\rvert\leqslant g(x,t)\left\lvert \nabla u\right\rvert^{\delta(x)} \] with \[ \delta (x)=\frac{p(x)(N+1)-N}{(N+2)(p(x)-1)}(p^--1) \quad \text{and} \quad p^-=\underset{x\in\bar{\Omega}}{min}\,p(x). \] By virtue of truncation metheod,the monotone operator theory and a gradient estimate we prove existence of renormalized solutions without coercivity condition on lower order term in the framework of variable exponents.