Non-delay limit in the energy space from the nonlinear damped wave equation to the nonlinear heat equation

TL;DR

This paper studies the singular limit from the damped wave equation to the heat equation, demonstrating improved convergence rates and global uniform convergence due to dissipation effects.

Contribution

It introduces the non-delay limit problem and provides enhanced convergence results compared to previous non-relativistic limit analyses.

Findings

Better convergence rates in the $L^2$-norm.

Global-in-time uniform convergence in the $L^2$-supercritical case.

Utilizes dissipation to improve limit results.

Abstract

We consider a singular limit problem from the damped wave equation with a power type nonlinearity to the corresponding heat equation. We call our singular limit problem non-delay limit. Our proofs are based on the argument for non-relativistic limit from the nonlinear Klein-Gordon equation to the nonlinear Schr\"{o}dinger equation by the second author, Nakanishi, and Ozawa (2002), Nakanishi (2002), and Masmoudi and Nakanishi (2002). We can obtain better results for the non-delay limit problem than that for the non-relativistic limit problem due to the dissipation property. More precisely, we get the better convergence rate of the $L^2$-norm and we also obtain the global-in-time uniform convergence of the non-delay limit in the $L^2$-supercritical case.