Affiliations: [a] IMAS-CONICET, Ciudad Universitaria, Buenos Aires, Argentina | [b] Departamento de Matemática, FCEyN-UBA, Ciudad Universitaria, Buenos Aires, Argentina
Correspondence:
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Corresponding author: Joana Terra, IMAS-CONICET, Ciudad Universitaria, Pabellón I (C1428EGA), Buenos Aires, Argentina. Tels: +5411 4576 3390, +5411 4576 3396, int. 901/902; Fax: +5411 4576 3335; E-mail: jterra@dm.uba.ar.
Abstract: In this paper we continue our study of the large time behavior of the bounded solution to the nonlocal diffusion equation with absorption
ut=Lu−upin RN×(0,∞),u(x,0)=u0(x)in RN,
where p>1, u0⩾0 and bounded and
Lu(x,t)=∫J(x−y)(u(y,t)−u(x,t))dy
with J∈C0∞(Bd), radially symmetric, J>0 in Bd, with ∫J=1. Our assumption on the initial datum is that 0⩽u0∈L∞(RN) and
|x|αu0(x)→A>0as |x|→∞.
This problem was studied in [Proc. Amer. Math. Soc. 139(4) (2011), 1421–1432; Discrete Cont. Dyn. Syst. A, 31(2) (2011), 581–605] in the supercritical and critical cases p⩾1+2/α. In the present paper we study the subcritical case 1<p<1+2/α. More generally, we consider bounded nonnegative initial data such that
|x|2/(p−1)u0(x)→∞as |x|→∞
and prove that
t1/(p−1)u(x,t)→(1p−1)1/(p−1)as t→∞
uniformly in |x|⩽kt for every k>0. Of independent interest is our study of the positive eigenfunction of the operator L in the ball BR in the L∞ setting that we include in Section 3.
