Decay estimates for nonlinear nonlocal diffusion problems in the whole space

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 $1\le p<\infty$. The upper and lower bounds that we obtain are sharp and provide an explicit expression for the first eigenvalue in the whole $\rr^d$: $$ λ_{1,p} (\rr^d) = 2(\int_{\rr^d} ψ(z) \, dz)|\frac{1}{|\det{A}|^{1/p}} -1|^p. $$ Moreover, we deal with the $p=\infty$ eigenvalue problem studying the limit as $p\to \infty$ of ${\lambda }_{1,p}^{1/p}$.

References in corpus (1)

• On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators