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- W4300695433 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 $lambda_{1,p}^{1/p}$." @default.
- W4300695433 created "2022-10-04" @default.
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- W4300695433 date "2012-07-11" @default.
- W4300695433 modified "2023-10-18" @default.
- W4300695433 title "Decay estimates for nonlinear nonlocal diffusion problems in the whole space" @default.
- W4300695433 doi "https://doi.org/10.48550/arxiv.1207.2565" @default.
- W4300695433 hasPublicationYear "2012" @default.
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