arXiv subjects

K. Khalil

Publications and source records attributed to K. Khalil.

5 recordsLinked to original sources

arXiv2023

Analysis of a spatio-temporal advection-diffusion model for human behaviors during a catastrophic event

In this work, using the theory of first-order macroscopic crowd models, we introduce a compartmental advection-diffusion model, describing the spatio-temporal dynamics of a population in different human behaviors (alert, panic and control) during a catastrophic event. For this model, we prove the local existence, uniqueness and regularity of a solution, as well as the positivity and $L^1$--boundedness of this solution. Then, in order to study the spatio-temporal propagation of these behavioral reactions within a population during a catastrophic event, we present several numerical simulations for different evacuation scenarios.

math.AP↗

arXiv2021

$({\mathrm R},{\mathcal B})$-Multidimensional-almost automorphic functions with applications to integral and partial differential equations

In this paper, we analyze the classes of $({\mathrm R},{\mathcal B})$-multi-almost automorphic functions and asymptotically $({\mathrm R},{\mathcal B})$-multi-almost automorphic functions. We provide plenty valuable applications to the abstract Volterra integrodifferential equations in Banach spaces and partial differential equations.

math.AP↗

arXiv2020

Multi-dimensional almost periodic type functions and applications

In this paper, we analyze multi-dimensional $({\mathrm R}_{X},{\mathcal B})$-almost periodic type functions and multi-dimensional Bohr ${\mathcal B}$-almost periodic type functions. The main structural characterizations and composition principles for the introduced classes of almost periodic functions are established. Several applications of our abstract theoretical results to the abstract Volterra integro-differential equations in Banach spaces are provided, as well.

math.FA↗

arXiv2020

Stepanov pseudo almost periodic functions and applications

In this work, we present basic results and applications of Stepanov pseudo almost periodic functions with measures. Using only the continuity assumption, we prove a new composition result of $\mu$-pseudo almost periodic functions in Stepanov sense. Moreover, we present different applications to semilinear differential equations and inclusions in Banach spaces with weak regular forcing terms. We prove the existence and uniqueness of $\mu$-pseudo almost periodic solutions (in the strong sense) to a class of semilinear fractional inclusions and semilinear evolution equations, respectively, provided that the nonlinear forcing terms are only Stepanov $ \mu $-pseudo almost periodic in the first variable and not a uniformly strict contraction with respect to the second argument. Some examples illustrating our theoretical results are also presented.

math.AP↗

arXiv2019

On the $L^p$-theory of vector-valued elliptic operators

In this paper, we study vector--valued elliptic operators of the form $\mathcal{L}f:=\mathrm{div}(Q\nabla f)-F\cdot\nabla f+\mathrm{div}(Cf)-Vf$ acting on vector-valued functions $f:\mathbb{R}^d\to\mathbb{R}^m$ and involving coupling at zero and first order terms. We prove that $\mathcal{L}$ admits realizations in $L^p(\mathbb{R}^d,\mathbb{R}^m)$, for $1<p<\infty$, that generate analytic strongly continuous semigroups provided that $V=(v_{ij})_{1\le i,j\le m}$ is a matrix potential with locally integrable entries satisfying a sectoriality condition, the diffusion matrix $Q$ is symmetric and uniformly elliptic and the drift coefficients $F=(F_{ij})_{1\le i,j\le m}$ and $C=(C_{ij})_{1\le i,j\le m}$ are such that $F_{ij},C_{ij}:\mathbb{R}^d\to\mathbb{R}^d$ are bounded. We also establish a result of local elliptic regularity for the operator $\mathcal{L}$, we investigate on the $L^p$-maximal domain of $\mathcal{L}$ and we characterize the positivity of the associated semigroup.

math.AP↗