Existence and Asymptotic Behavior of Solutions to Some Inhomogeneous Nonlocal Diffusion Problems
Carmen Cortázar, Manuel Elgueta, Jorge García-Melián, Salomé Martínez · SIAM Journal on Mathematical Analysis · 2009
We consider the nonlocal evolution Dirichlet problem $u_t(x,t)=\int_{\Omega}J(\frac{x-y}{g(y)})\frac{u(y,t)}{g(y)^N}dy-u(x,t)$, $x\in\Omega$, $t>0$; $u=0$, $x\in\mathbb{R}^N\setminus\Omega$, $t\ge0$; $u(x,0)=u_0(x)$, $x\in\mathbb{R}^N$; where $\Omega$ is a bounded domain in $\mathbb{R}^N$, J is a Hölder continuous, nonnegative, compactly supported function with unit integral and $g\in C(\overline{\Omega})$ is assumed to be positive in $\Omega$. We discuss existence, uniqueness, and asymptotic behavior of solutions as $t\to+\infty$. Moreover, we prove the existence of a positive stationary solution when the inequality $g(x)\le\delta(x)$ holds at every point of $\Omega$, where $\delta(x)=\mathrm{dist}(x,\partial\Omega)$. The behavior of positive stationary solutions near the boundary is also analyzed.