Generalized diffusion problems in a conical domain, part I

TL;DR

This paper investigates the behavior of solutions to a reaction-diffusion equation with a generalized dispersal operator in a conical domain, focusing on the solution's behavior near the cone's apex within weighted Sobolev spaces.

Contribution

It provides a detailed analysis of the solution's behavior near the cone's apex in weighted Sobolev spaces, extending understanding of diffusion problems in conical geometries.

Findings

Solution behavior near the cone's top is fully characterized in weighted Sobolev space $W^{4,p}_{3-rac{1}{p}}(S_{ ho, heta})$

The study offers a rigorous description of the dispersal operator in conical domains

The analysis enhances mathematical understanding of reaction-diffusion equations in non-smooth geometries.

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 $W^{4,p}_{3-\frac{1}{p}}(S_{\omega,\rho})$, see Theorem 2.2.