Stability for evolution equations with variable growth

TL;DR

This paper investigates the stability of solutions to a variable exponent p(x,t)-Laplacian evolution equation, providing quantitative estimates on how solutions depend continuously on data, nonlinearities, and variable exponents.

Contribution

It establishes the stability of solutions with respect to perturbations in the variable exponent, source term, and initial data, including explicit convergence rates.

Findings

Solutions are stable under perturbations of p(x,t), f, and initial data.

Quantitative estimates relate solution differences to data perturbations.

Convergence rates for solution sequences are derived.

Abstract

We study the character of dependence on the data and the nonlinear structure of the equation for the solutions of the homogeneous Dirichlet problem for the evolution $p(x,t)$-Laplacian with the nonlinear source \[ u_t-\Delta_{p(x,t)}u=f(x,t,u),\quad (x,t)\in Q=\Omega\times (0,T), \] where $\Omega$ is a bounded domain in $\mathbb{R}^n$, $n\geq 2$, and $p(x,t)$ is a given function $p(\cdot):Q\mapsto (\frac{2n}{n+2},p^+]$, $p^+<\infty$. It is shown that the solution is stable with respect to perturbations of the variable exponent $p(x,t)$, the nonlinear source term $f(x,t,u)$, and the initial data. We obtain quantitative estimates on the norm of the difference between two solutions in a variable Sobolev space through the norms of perturbations of the nonlinearity exponent and the data $u(x,0)$, $f$. Estimates on the rate of convergence of a sequence of solutions to the solution of the limit problem are derived.