Harnack inequality and self-similar solutions for fully nonlinear fractional parabolic equations
The paper is devoted to the nonlocal eigenvalue problem \[ \mathcal{M}\Phi+\frac1{2s}\,x\cdot D\Phi=-\lambda\Phi\text{ in } \mathbb{R}^n, \] which is motivated by the connection with self-similar solutions for the parabolic problem \[ u_t=\mathcal{M} u \text{ in }\mathbb{R}^n \times \mathbb{R}_+. \] The authors state the results (Theorem 1.1) in the case that \(\mathcal{M}\) is a nonlocal extremal Pucci operator, but it has to be remarked that they can be also extended to non-convex nonlinearities. Precisely, in the case \(s\in (1/2,1)\) and kernel of order \(|x|^{-(N+2s)}\), the authors show the existence of an eigenpair \((\Phi_1^+,\lambda_1^+)\), where the eigenfunction \(\Phi_1^+>0\) has the same polynomial decay rate as the kernel, and the eigenvalue \(\lambda_1^+>0\) is simple and unique in a suitable sense. The existence is obtained by approximation on bounded domains, where the convergence is based on a Harnack inequality (Theorem 1.3), established quite generally in the nonlocal case with zero-order terms. The lower and upper bounds for the eigenvalue and the decay estimates for the eigenfunction rely on the fundamental solutions for the nonlocal operator \(\mathcal{M}\). The existence of positive self-similar solutions for the related parabolic problem is also proved with an explicit representation in terms of \(\Phi_1^+\) (Corollary 1.2).
decay estimates for the principal eigenfunction
scientific article; zbMATH DE number 7613196
scientific article