Asymptotic behavior of the nonlocal diffusion equation with localized source

Abstract:

In this paper we study the nonlocal diffusion equation with localized source: $u_t= J*u-u+a(x)u^p$ in $\mathbb {R}^N\times (0,T)$, with $a(x)$ nonnegative, continuous, and compactly supported. It is found that the localized source $a(x)$ drastically changes the asymptotic behavior of the nonlocal diffusion equation that the Fujita phenomenon happens only if $N=1$. That is to say, the solutions must be global provided the initial data are small if $N>1$. Furthermore, we determine the second critical exponent $b_c=\frac {1}{p-1}$ with $N=1$, and $b_c=0$ with $N>1$, rather than $b_c=\frac {2}{p-1}$ for the case of homogeneous source with all $N\ge 1$. This implies that the scope of initial data for global solutions determined by the second critical exponent $b_c$ is enlarged due to the localized factor $a(x)$. Finally, the time-decay profile of the global solutions is also studied for slow-decay initial data. It is mentioned that we need some new techniques to deal with the nonlocal diffusion in the model. For example, different from local diffusion equations, because of a lack of regularity mechanism from the nonlocal diffusion, we employ the moving plane method in integral form to deal with the mild solutions instead of the maximum principle. In addition, due to the localization of the source, we have to use precise weighted $L^1$ estimates for the critical situation to replace the general test function method.