Decay estimates for nonlinear nonlocal diffusion problems in the whole space

TL;DR

This paper derives sharp decay estimates for solutions to a nonlinear nonlocal diffusion equation in the whole space, explicitly characterizing the first eigenvalue of the associated operator for various p-values.

Contribution

It provides explicit formulas and bounds for the first eigenvalue of a nonlocal diffusion operator with a specific kernel, extending understanding of decay rates in nonlinear nonlocal problems.

Findings

Explicit expression for the first eigenvalue in  space

Sharp upper and lower bounds for eigenvalues

Analysis of the limit as p approaches infinity

Abstract

In this paper we obtain bounds for the decay rate in the $L^r (\rr^d)$-norm for the solutions to a nonlocal and nolinear evolution equation, namely, $$u_t(x,t) = \int_{\rr^d} K(x,y) |u(y,t)- u(x,t)|^{p-2} (u(y,t)- u(x,t)) \, dy, $$ with $ x \in \rr^d$, $ t>0$. Here we consider a kernel $K(x,y)$ of the form $K(x,y)=\psi (y-a(x))+\psi(x-a(y))$, where $\psi$ is a bounded, nonnegative function supported in the unit ball and $a$ is a linear function $a(x)= Ax$. To obtain the decay rates we derive lower and upper bounds for the first eigenvalue of a nonlocal diffusion operator of the form $ T(u) = - \int_{\rr^d} K(x,y) |u(y)-u(x)|^{p-2} (u(y)-u(x)) \, dy$, with $1 \leq p < \infty$. The upper and lower bounds that we obtain are sharp and provide an explicit expression for the first eigenvalue in the whole $\rr^d$: $$ \lambda_{1,p} (\rr^d) = 2(\int_{\rr^d} \psi (z) \, dz)|\frac{1}{|\det{A}|^{1/p}} -1|^p. $$ Moreover, we deal with the $p=\infty$ eigenvalue problem studying the limit as $p \to \infty$ of $\lambda_{1,p}^{1/p}$.