Solvability of a transmission problem in $L^p$-spaces with generalized diffusion equation

TL;DR

This paper investigates the solvability of a transmission problem in population dynamics involving generalized diffusion equations with Laplace and biharmonic operators, establishing conditions for existence and uniqueness of solutions in $L^p$-spaces.

Contribution

It introduces new conditions on coefficients for the existence and uniqueness of solutions to a complex diffusion transmission problem using semigroup theory.

Findings

Derived relations between coefficients for solution existence

Proved uniqueness of classical solutions in $L^p$-spaces

Established conditions for solvability of generalized diffusion equations

Abstract

We study a transmission problem, in population dynamics, between two juxtaposed habitats. In each habitat, we consider a generalized diffusion equation composed by the Laplace operator and a biharmonic term. We consider that the coefficients in front of each term could be negative or null. Using semigroups theory and functional calculus, we give some relation between coefficients to obtain the existence and the uniqueness of the classical solution in $L^p$-spaces.