ASYMPTOTIC BEHAVIOR FOR A NONLOCAL DIFFUSION EQUATION ON THE HALF LINE

C. CORTAZAR, M. ELGUETA, F. QUIROS, N. WOLANSKI

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS

AMER INST MATHEMATICAL SCIENCES

Lugar: springfield; Año: 2015 vol. 35 p. 1391 - 1407

1078-0947

We study the large time behavior of solutions to anonlocal diffusion equation, $u_t=J*u-u$ with $J$ smooth, radially symmetric and compactly supported, posed in $\mathbb{R}_+$ with zero Dirichlet boundary conditions. In the far-field scale, $\xi_1\le xt^{-1/2}\le \xi_2$ with $\xi_1,\xi_2>0$, the asymptotic behavior is given by a multiple of the dipole solution for the local heat equation, hence $tu(x,t)$ is bounded above and below by positive constants in this region for large times. The proportionality constant is determined from a conservation law, related to the asymptotic first momentum. In compact sets, after scaling the solution by a factor $t^{3/2}$, it converges to a multiple of the unique stationary solution of the problem that behaves as $x$ at infinity. The precise proportionality factor is obtained through a matching procedure with the far-field limit. Finally, in the very far-field, $x\ge t^{1/2} g(t)$ with $g(t)\to\infty$, the solution is proved to be of order $o(t^{-1})$.