arXiv subjects

Maha Daoud

Publications and source records attributed to Maha Daoud.

5 recordsLinked to original sources

arXiv2026

On some 1D nonlocal models with coefficients changing sign

In this work, we study one-dimensional nonlocal elliptic transmission problems with piecewise constant coefficients that may change sign across an interface. In the local setting, we recall the T-coercive structure of the problem and characterize the critical contrast case. In the nonlocal setting, we focus on a simplified configuration in which the cross-interaction coefficient vanishes. Under this assumption, we prove a weak T-coercivity result for the global fractional problem and introduce a reconstructed formulation based on an explicit interface lifting. Then, we consider a simplified finite element discretization of the reconstructed model and prove its convergence toward the classical local transmission problem as the fractional parameter $s\to 1^-$ and the mesh size $h\to 0^+$. Numerical simulations in 1D illustrate the stability and consistency of the method, and a preliminary two-dimensional extension is presented as an exploratory perspective.

math.AP↗

arXiv2026

Lower and Upper Bounds for Sums of Eigenvalues of the Fractional-Logarithmic Laplacian

In this work, we establish lower and upper bounds for sums of the Dirichlet eigenvalues of the fractional-logarithmic Laplacian. A main challenge in such a study comes from the fact that this operator has a Fourier symbol that is not globally monotone in its radial variable due to its low-frequency behavior. The bounds have the correct leading-order asymptotic. To complement the theoretical analysis, we also present a one-dimensional finite element approximation of the fractional-logarithmic eigenvalue problem. We compare the computed eigenvalue sums with the explicit lower bound and the principal asymptotic term

math.AP↗

arXiv2025

Fractional elliptic reaction-diffusion systems with coupled gradient terms and different diffusion

In this work, we study the existence and nonexistence of nonnegative solutions to a class of nonlocal elliptic systems set in a bounded open subset of $\mathbb{R}^N$. The diffusion operators are of type $u_i\mapsto d_i(-\Delta)^{s_i}u_i$ where $0<s_1\neq s_2<1$, and the gradients of the unknowns act as source terms. Existence results are obtained by proving some fine estimates when data belong to weighted Lebesgue spaces. Those estimates are new and interesting in themselves.

math.AP↗

arXiv2025

A class of parabolic reaction-diffusion systems governed by spectral fractional Laplacians : Analysis and numerical simulations

In this paper, we prove the global-in-time existence of strong solutions to a class of fractional parabolic reaction-diffusion systems set in a bounded open subset of $\mathbb{R}^N$. The diffusion operators are of the form $u_i \mapsto d_i (-\Delta)_{Sp}^{s_i} u_i$, where $0 < s_i < 1$. The operator $(-\Delta)_{Sp}^{s}$ stands for the commonly called spectral fractional Laplacian. Moreover, the nonlinear reaction terms are assumed to fulfill natural structural conditions that ensure the nonnegativity of the solutions and provide uniform control of the total mass. We establish the global existence of strong solutions under the assumption that the nonlinearities exhibit at most polynomial growth. Our results extend previous results obtained when the diffusion operators are of the form $u_i \mapsto d_i (-\Delta)^s u_i$, where $(-\Delta)^s$ denotes the widely known regional fractional Laplacian. Furthermore, we present some numerical simulations to address a theoretical question that remains open to date.

math.AP↗

arXiv2023

A class of fractional parabolic reaction-diffusion systems with control of total mass: theory and numerics

In this paper, we prove global-in-time existence of strong solutions to a class of fractional parabolic reaction-diffusion systems posed in a bounded domain of $\mathbb{R}^N$. The nonlinear reactive terms are assumed to satisfy natural structure conditions which provide non-negativity of the solutions and uniform control of the total mass. The diffusion operators are of type $u_i\mapsto d_i(-\Delta)^s u_i$ where $0<s<1$. Global existence of strong solutions is proved under the assumption that the nonlinearities are at most of polynomial growth. Our results extend previous results obtained when the diffusion operators are of type $u_i\mapsto -d_i\Delta u_i$. On the other hand, we use numerical simulations to examine the global existence of solutions to systems with exponentially growing right-hand sides, which remains so far an open theoretical question even in the case $s=1$.

math.AP↗