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

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 10. 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.