Large time behavior for a nonlocal diffusion equation with absorption and bounded initial data: The subcritical case

Article type: Research Article

Authors: Salort, Ariel^{a; b} | Terra, Joana^{a; b; *} | Wolanski, Noemi^{a; b}

Affiliations: [a] IMAS-CONICET, Ciudad Universitaria, Buenos Aires, Argentina | [b] Departamento de Matemática, FCEyN-UBA, Ciudad Universitaria, Buenos Aires, Argentina

Correspondence: [*] 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: [email protected].

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.

Keywords: nonlocal diffusion, large time behavior

DOI: 10.3233/ASY-151320

Journal: Asymptotic Analysis, vol. 95, no. 1-2, pp. 39-57, 2015