Generalized diffusion problems in a conical domain, part I

Abstract

The purpose of this article (composed of two parts) is the study of the generalized dispersal operator of a reaction-diffusion equation in $L^p$-spaces set in the finite conical domain $S_{\omega ,\rho }$ of angle $\omega >0$ and radius $\rho >0$ in ${\mathbb{R}}^2$. This first part is devoted to the behaviour of the solution near the top of the cone which is completely described in the weighted Sobolev space $W_{3-\frac{1}{p}}^{4,p}(S_{\omega ,\rho })$, see Theorem 2.2.