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This paper analyzes positive solutions to fully nonlinear nonlocal diffusion equations with a vanishing coefficient $k(x)$ on a sanctuary region. It characterizes the existence interval via principal half-eigenvalues and proves uniqueness, showing that at the lower endpoint, solutions vanish uniformly and converge to the principal eigenfunction. Crucially, as the diffusion parameter $\mu$ approaches a critical value related to the sanctuary region, solutions exhibit a two-scale blow-up, converging to the sanctuary's principal eigenfunction on one scale and a rescaled version of the local diffusion term on another.
Two-scale blow-up emerges in nonlocal diffusion equations when coefficients vanish, revealing distinct convergence behaviors for solutions.
We study positive solutions of $Iu+\mu u=k(x)u^p$ in a bounded domain, where $p>1$, the nonnegative coefficient $k$ may vanish on a sanctuary region, and $I$ is a fully nonlinear nonlocal operator uniformly elliptic in the Caffarelli--Silvestre sense. We characterize the sharp existence interval in terms of positive principal half-eigenvalues and prove uniqueness. At the lower endpoint, the solutions vanish uniformly and their normalized profiles converge to the principal eigenfunction. If $k$ vanishes on a sanctuary $\Omega_0$, then, as $\mu\uparrow\lambda(\Omega_0)$, the solutions blow up locally uniformly in $\Omega$. Writing $\phi_0=\phi(\cdot,\Omega_0)$ for the normalized positive principal eigenfunction in $\Omega_0$, we prove \[ \frac{u_\mu}{\|u_\mu\|_{L^\infty(\Omega)}}\to\phi_0, \qquad \frac{u_\mu}{\|u_\mu\|_{L^\infty(\Omega)}^{1/p}} \to\left(\frac{I\phi_0}{k}\right)^{1/p}, \] uniformly in $\mathbb{R}^n$ and locally uniformly in $\Omega\setminus\overline{\Omega_0}$, respectively. Thus the upper-endpoint blow-up occurs on two distinct scales.