Generalized linear models for population dynamics in two juxtaposed habitats

Abstract

In this work we introduce a generalized linear model regulating the spread of population displayed in a d-dimensional spatial region Ω of R d constituted by two juxtaposed habitats having a common interface Γ. This model is described by an operator L of fourth order combining the Laplace and Biharmonic operators under some natural boundary and transmission conditions. We then invert explicitly this operator in L p-spaces using the H ∞-calculus and the Dore-Venni sums theory. This main result will lead us in a later work to study the nature of the semigroup generated by L which is important for the study of the complete nonlinear generalized diffusion equation associated to it.

A biharmonic transmission problem in <i>L^p</i>-spaces

“…This work is a natural continuation of that done in [31]. Moreover, it completes the study realized in [20] where the authors have considered equations…”

“…Here, we refer the reader for instance to [1,9,15] for applications in plate theory, to [18,26,34] for applications in electromagnetism and to [10,21,33] for other applications in population dynamics. Let us also mention that mathematical models involving biharmonic operators also arise in various fields such as elasticity for instance see [6,16,30], electrostatic see [4,13,22], plate theory [1,9,15] or population dynamics [5,19,20,25].…”

Operational approach for biharmonic equations in $${\varvec{L}}^{{\varvec{p}}}$$-spaces

“…4p ,p , which leads to (18) by using the reiteration theorem and Theorem in section 1.14.3 in [33]. Noting that A = M 2 , again by the reiteration theorem and Theorem in section 1.14.3 in [33], we obtain (19).…”

“…One may also refer to [3], [14] or [23] for biharmonic models in electrostatic, to [8], [11] or [32] for applications in plate theory where various different boundary conditions arise. We also refer to [4], [19], [20] or [26] for models in population dynamics.…”

On a generalized diffusion problem: A complex network approach

“…Many theoretical results of existence, uniqueness and regularity have been obtained for fourth order problems like Cahn-Hilliard equation, see for instance [10,17,19,33,36]. More recently, problem (1) has been studied in a general Banach space setting for the linear stationary case [28,29,46]. Here, we aim to improve and extend the results of the latter papers to the non-stationary case.…”