Decay rates to equilibrium in a nonlinear subdiffusion equation with two counteracting terms

TL;DR

This paper establishes the convergence to equilibrium for a class of nonlinear subdiffusion equations with two opposing terms, demonstrating decay rates under broad conditions.

Contribution

It proves decay to steady state for nonlinear subdiffusion equations with general coefficients and nonlinearities, covering both classical and fractional cases.

Findings

Convergence to steady state as t→∞

Decay rates are exponential for α=1 and power law for α<1

Results hold under mild conditions on coefficients and nonlinearities

Abstract

In this paper we prove convergence to a steady state as $t\to\infty$ for solutions to the subdiffusion equation \[ \partial_t^\alpha u - \mathbb{L} u = q(x)u - p(x)f(u) + r \] with the exponential ($\alpha=1$) or power law ($\alpha\in[0,1)$) rates under mild conditions on the coefficients $p$, $q$, the nonlinearity $f$, the source $r$, and the elliptic operator $\mathbb{L}$.