In this paper, we consider a generalized diffusion problem arising in population dynamics. To this end, we study a fourth order operational equation of elliptic type, with various boundary conditions. We show existence, uniqueness and regularity of a classical solution on a cylindrical domain under some necessary and sufficient conditions on the data. This elliptic problem is solved in L p (a, b; X), p ∈ (1, +∞), where (a, b) ⊂ R and X is a UMD Banach space. Our techniques use essentially the functional calculus and the semigroup theory.

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.

In this work, we study the existence, uniqueness and maximal L p-regularity of the solution of different biharmonic problems. We rewrite these problems by a fourth order operational equation and different boundary conditions, set in a cylindrical n-dimensional spatial region Ω of R n. To this end, we give an explicit representation formula, using analytic semigroups, and invert explicitly a determinant operator in L p-spaces thanks to E ∞ functional calculus and operator sums theory.

In this paper, we propose a new approach for studying a generalized diffusion problem, using complex networks of reaction-diffusion equations. We model the biharmonic operator by a network, based on a finite graph, in which the couplings between nodes are linear. To this end, we study the generalized diffusion problem, establishing results of existence, uniqueness and maximal regularity of the solution via operator sums theory and analytic semigroups techniques. We then solve the complex network problem and present sufficient conditions for the solutions of both problems to converge to each other. Finally, we analyze their asymptotic behavior by establishing the existence of a family of exponential attractors.

We study the solvability of some boundary-value problems for differential-operator equations of the second order in L p (0, 1; X), with 1 < p < +∞, X being a UMD complex Banach space. The originality of this work lies in the fact that we consider the case where two spectral complex parameters appear in the equation and in abstract Robin boundary conditions. Here, the unbounded linear operator in the equation is not commuting with the one appearing in the boundary conditions. This represents the strong novelty with respect to the existing literature. Existence, uniqueness, representation formula, maximal regularity of the solution, sharp estimates and generation of strongly continuous analytic semigroup are proved. Many concrete applications are given for which our theory applies. This paper improves, in some sense, results by the authors in [7] and it can be viewed as a continuation of the results in [1] studied only in Hilbert spaces.