Homogenization of a semilinear heat equation

TL;DR

This paper studies the homogenization process of a semilinear heat equation with oscillating potential and vanishing viscosity, revealing different limiting behaviors depending on oscillation and diffusion regimes, including a discontinuous effective operator in strong diffusion cases.

Contribution

It provides a complete characterization of the limit solution in one dimension and analyzes the properties and uniqueness of solutions in higher dimensions.

Findings

Different regimes depending on oscillation and viscosity rates.

Discontinuous effective operator in strong diffusion regime.

Complete characterization of limit solutions in 1D.

Abstract

We consider the homogenization of a semilinear heat equation with vanishing viscosity and with oscillating positive potential depending on $u/\varepsilon$. According to the rate between the frequency of oscillations in the potential and the vanishing factor in the viscosity, we obtain different regimes in the limit evolution and we discuss the locally uniform convergence of the solutions to the effective problem. The interesting feature of the model is that in the strong diffusion regime the effective operator is discontinuous in the gradient entry. We get a complete characterization of the limit solution in dimension $n=1$, whereas in dimension $n>1$ we discuss the main properties of the solutions to the effective problem selected at the limit and we prove uniqueness for some classes of initial data.