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 $$\left\{\begin{array}{cc}{\mathit{u}}_{\mathit{t}}=\mathcal{L}\mathit{u}-{\mathit{u}}^{\mathit{p}}\hfill & \text{in }{\mathbb{R}}^{\mathit{N}}\times \mathrm{(}0\mathrm{,}\infty \mathrm{)}\mathrm{,}\hfill \\ \mathit{u}\mathrm{(}\mathit{x}\mathrm{,}0\mathrm{)}={\mathit{u}}_0\mathrm{(}\mathit{x}\mathrm{)}\hfill & \text{in }{\mathbb{R}}^{\mathit{N}}\mathrm{,}\hfill \end{array}\right.$$ where $\mathit{p}\mathrm{>}1$ , ${\mathit{u}}_0⩾0$ and bounded and $$\mathcal{L}\mathit{u}\mathrm{(}\mathit{x}\mathrm{,}\mathit{t}\mathrm{)}={\displaystyle \int }\mathit{J}\mathrm{(}\mathit{x}-\mathit{y}\mathrm{)}\mathrm{(}\mathit{u}\mathrm{(}\mathit{y}\mathrm{,}\mathit{t}\mathrm{)}-\mathit{u}\mathrm{(}\mathit{x}\mathrm{,}\mathit{t}\mathrm{)}\mathrm{)}\mathrm{d}\mathit{y}$$ with $\mathit{J}\in {\mathit{C}}_0^{\infty }\mathrm{(}{\mathit{B}}_{\mathit{d}}\mathrm{)}$ , radially symmetric, $\mathit{J}\mathrm{>}0$ in ${\mathit{B}}_{\mathit{d}}$ , with $\int \mathit{J}=1$ .

Our assumption on the initial datum is that $0⩽{\mathit{u}}_0\in {\mathit{L}}^{\infty }\mathrm{(}{\mathbb{R}}^{\mathit{N}}\mathrm{)}$ and $$|\mathit{x}{|}^{\mathit{\alpha }}{\mathit{u}}_0\mathrm{(}\mathit{x}\mathrm{)}\to \mathit{A}\mathrm{>}0\text{as }|\mathit{x}|\to \infty .$$

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 $\mathit{p}⩾1+2\mathrm{/}\mathit{\alpha }$ .

In the present paper we study the subcritical case $1\mathrm{<}\mathit{p}\mathrm{<}1+2\mathrm{/}\mathit{\alpha }$ . More generally, we consider bounded nonnegative initial data such that $$|\mathit{x}{|}^{2\mathrm{/}\mathrm{(}\mathit{p}-1\mathrm{)}}{\mathit{u}}_0\mathrm{(}\mathit{x}\mathrm{)}\to \infty \text{as }|\mathit{x}|\to \infty$$ and prove that $${\mathit{t}}^{1\mathrm{/}\mathrm{(}\mathit{p}-1\mathrm{)}}\mathit{u}\mathrm{(}\mathit{x}\mathrm{,}\mathit{t}\mathrm{)}\to {\mathrm{\left(}{\displaystyle \frac{1}{\mathit{p}-1}}\mathrm{\right)}}^{1\mathrm{/}\mathrm{(}\mathit{p}-1\mathrm{)}}\text{as }\mathit{t}\to \infty$$ uniformly in $|\mathit{x}|⩽\mathit{k}\sqrt{\mathit{t}}$ for every $\mathit{k}\mathrm{>}0$ .

Of independent interest is our study of the positive eigenfunction of the operator $\mathcal{L}$ in the ball ${\mathit{B}}_{\mathit{R}}$ in the ${\mathit{L}}^{\infty }$ setting that we include in Section 3.