Decay estimates for nonlinear nonlocal diffusion problems in the whole space

In this paper we obtain bounds for the decay rate in the $L^r({\text{rr}}^d)$-norm for the solutions to a nonlocal and nolinear evolution equation, namely, $$u_t(x,t)={\int }_{{\text{rr}}^d}K(x,y)\mathrm{\mid }u(y,t)-u(x,t){\mathrm{\mid }}^{p-2}(u(y,t)-u(x,t))dy,$$ with $x\in {\text{rr}}^d$, $t>0$. 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 ${\text{rr}}^d$: $${\lambda }_{1,p}({\text{rr}}^d)=2({\int }_{{\text{rr}}^d}\psi (z)dz)\mathrm{\mid }\frac{1}{\mathrm{\mid }\det A{\mathrm{\mid }}^{1\mathrm{/}p}}-1{\mathrm{\mid }}^p\mathrm{.}$$ Moreover, we deal with the $p=\mathrm{\infty }$ eigenvalue problem studying the limit as $p\to \mathrm{\infty }$ of ${\lambda }_{1,p}^{1\mathrm{/}p}$.