Asymptotic behavior for a nonlocal diffusion problem with Neumann boundary conditions and a reaction term


In this paper, we consider the following initial value problem u_t(x,t)=\int_{\Omega}J(x-y)(u(y,t)-u(x,t))dy-\gamma u^{p}(x,t)& \mbox{in}& \overline{\Omega}\times(0,\infty),
u(x,0)=u_{0}(x)>0& \mbox{in}& \overline{\Omega},            where \gamma\ in \{-1,1\} is a parameter, \Omega is a bounded domain in      \mathbb{R}^{N} with smooth boundary \partial\Omega, p>1, J:      \mathbb{R}^N\longrightarrow\mathbb{R} is a kernel which is nonnegative,      measurable, symmetric, bounded and \int_{\mathbb{R}^N}J(z)dz=1, the      initial datum u_0 \ in C^0(\overline{\Omega}), u_0(x)>0 in      \overline{\Omega}. We show that, if \gamma=1, then the solution u      of the above problem tends to zero as t\rightarrow\infty uniformly in      x\in\overline{\Omega}, and a description of its asymptotic behavior is      given. We also prove that, if \gamma=-1, then the solution u blows up      in a finite time, and its blow-up time goes to that of the solution of a      certain ODE as the L^{\infty} norm of the initial datum goes to      infinity.

DOI Code: 10.1285/i15900932v30n1p1

Nonlocal diffusion; asymptotic behavior; blow-up time

Classification: 35B40; 45A07; 35G10

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