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Mohamed Majdoub

Publications and source records attributed to Mohamed Majdoub.

At least 19 recordsLinked to original sources
arXiv2026

Global existence and optimal decay for a three-dimensional penalized Navier--Stokes system with biharmonic damping

We investigate a three-dimensional parabolic system that arises as a hyperviscous and penalized approximation of the incompressible Navier--Stokes equations. The model combines three complementary dissipative mechanisms: the classical viscous diffusion, a biharmonic (hyperviscous) regularization, and a divergence penalization. In addition, a Temam-type correction is incorporated into the nonlinear convection term to compensate for the weak compressibility effects generated by the penalization procedure. We prove the global existence of weak solutions for arbitrary initial data belonging to $L^2(\mathbb{R}^3)$. For sufficiently small initial data in $H^2(\mathbb{R}^3)$, we establish the existence and uniqueness of global strong solutions. Furthermore, for initial data in $L^1(\mathbb{R}^3)\cap H^2(\mathbb{R}^3)$, we derive optimal large-time decay estimates, showing that the solutions exhibit the same asymptotic decay rates as those of the classical heat equation. A key feature of our analysis is that all the obtained a priori estimates are uniform with respect to the positive penalization parameter $\varepsilon$. These uniform bounds provide a stable and rigorous analytical foundation for the study of the penalized approximation of incompressible flows.

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arXiv2026

Well-Posedness of a Coupled Brinkman--Biofilm--Nutrient System with Volume-Fraction Constraints

We investigate a coupled system of partial differential equations modeling the interaction between Brinkman flow, biofilm evolution, and nutrient transport in a porous medium. The model captures the mutual influence between the fluid velocity and the biofilm through drag and diffusion coefficients that depend on the local biofilm volume fraction. A hard constraint on the admissible range of the biofilm fraction is incorporated through the subdifferential of an indicator functional, which leads naturally to an evolution variational inequality formulation for the biofilm dynamics. Assuming standard coercivity, ellipticity, and growth conditions on the model coefficients and reaction terms, we prove the global-in-time existence of weak solutions. The analysis relies on a decomposition of the system into three interconnected subproblems: the Brinkman equation with a fixed biofilm profile, the constrained biofilm evolution treated through maximal monotone operator theory, and the nutrient equation viewed as a semilinear parabolic problem. These components are then coupled through a Leray--Schauder type fixed-point argument, with the passage to the limit justified by Aubin--Lions and Simon compactness results. We further establish the nonnegativity of the nutrient concentration under a natural quasi-positivity assumption on the reaction term. Finally, we provide conditional uniqueness results for weak solutions in two spatial dimensions under additional smallness assumptions.

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arXiv2026

Fujita-type blow-up for inhomogeneous semilinear heat equations with regularly varying forcing

We develop a unified framework for Fujita-type blow-up of solutions to the inhomogeneous semilinear heat equation $$\partial_tu-Δu=|u|^p+\mathbf{w}(x), \qquad (t,x)\in(0,\infty)\times\mathbb{R}^N, \qquad u(0, \cdot)=u_0.$$ The classical integrability assumptions on the forcing term are replaced by quantitative regular variation properties of its spatial mass $$F(R)=\int\limits_{|x|\le R}\mathbf{w}(x)\,dx.$$ Using techniques from regular variation theory together with the Mitidieri--Pohozaev test-function method, we establish sharp Fujita-type nonexistence results and identify the critical exponent in terms of the variation index of $F$. We prove that global solutions do not exist in the subcritical range and obtain critical-case blow-up under suitable slowly varying corrections. The regular variation framework further shows the optimality of the underlying mass condition, extends naturally to anisotropic settings through operator regular variation, and yields sufficient blow-up criteria for sign-changing forcings via the Gaussian-Laplace transform. The approach also applies to space-time dependent forcings, Riesz-potential type forcings, and equations involving the fractional Laplacian, providing a unified description of blow-up thresholds beyond the classical Fujita theory.

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arXiv2026

The Fujita exponent across an interface

We consider the semilinear parabolic equation \[ \partial_t u = Δu + 2\mathfrak{q}\,δ_{\mathbb{S}}\,\nabla u + |u|^{p-1}u \qquad \text{in } (0,\infty)\times\mathbb{R}^N, \] where $|\mathfrak{q}|\le 1$, $p>1$, and $\mathbb{S}$ is a fixed interface hyperplane. Working in Lebesgue spaces, we first establish local well-posedness of mild solutions. This is achieved by combining Gaussian bounds for the associated fundamental solution with a contraction mapping argument adapted to the lack of spatial homogeneity induced by the interface term. We then prove a sharp Fujita-type dichotomy for nonnegative solutions. Specifically, we show that every nontrivial solution blows up in finite time when $1 1+\frac{2}{N}$ global solutions exist for sufficiently small initial data. The blow-up analysis relies on a suitably adapted test-function method that accounts for the presence of the interface. It is noteworthy that the critical exponent coincides with the classical Fujita exponent for the heat equation, indicating that the Fujita phenomenon remains stable under the presence of discontinuous diffusion effects and interface transmission conditions. To the best of our knowledge, this is the first result of this type for operators involving a singular drift supported on a hypersurface.

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arXiv2026

Sharp Lifespan Estimates and Fujita Phenomena for Fractional Hardy-Hénon Type Parabolic Equations

We study the lifespan of mild solutions to the fractional semilinear parabolic Cauchy problem with a Hardy--Hénon-type weight \[ u_t + (-Δ)^s u = |x|^{-γ}\,|u|^p, \qquad (t,x)\in(0,\infty)\times\mathbb{R}^N, \qquad u(0,x)=\varepsilon\,u_0(x), \] where $0 1$ and $u_0\in L^1\cap L^\infty$ with $\int_{\mathbb{R}^N}u_0(x)\,dx>0$. Setting \[ p_F \;:=\; 1+\frac{2s-γ}{N}, \] we prove that the lifespan $T_\varepsilon$ obeys, for every sufficiently small $\varepsilon>0$, \[ T_\varepsilon \;\approx\; \begin{cases} \varepsilon^{-\,β^{-1}},& 1 p_F, \end{cases} \qquad β\;=\;\frac{(2s-γ)-N(p-1)}{2s(p-1)}. \] The lower bound rests on fractional heat-kernel estimates and an $L^1$--$L^\infty$ Hardy-type interpolation inequality; the upper bound is obtained by testing the equation against the backward fractional heat kernel, a globally defined positive weight for which $(-Δ)^s$ is controlled everywhere and the linear terms cancel identically by self-adjointness. This circumvents the compactly supported cutoffs of the classical test-function method, which are incompatible with a nonlocal operator. The exponent $β$ is sharp; for $γ=0$ it reduces to the fractional Lee--Ni exponent $\frac{1}{p-1}-\frac{N}{2s}$. To the best of our knowledge, these results are new even for $γ=0.$ We also establish a large-data lifespan law, sharp lower bounds on the blow-up rate together with a conditional Type-I upper bound, a conditional self-similar profile result.

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arXiv2026

An extension of a critical Hardy--Rellich inequality: explicit constants and the sharp weight range

We revisit the critical Hardy--Rellich inequalities recently established by Castro in "A critical Hardy--Rellich inequality (preprint, arXiv:2511.16537, 2025)". Using the classical one-dimensional weighted Hardy inequality in Emden--Fowler variables, we prove that the weighted inequality holds exactly for $a\neq N$, with $a=N$ as the unique critical value. We derive explicit constants for $N\ge2$, obtain the sharp constant in dimension one, and extend the $Δu$ formulation to the Muckenhoupt range $-N<a<N(N-1)$, $a\neq N$. This provides a partial solution to an open problem raised in Castro's work.

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arXiv2026

Fujita Phenomenon for a Mixed Local-Nonlocal Hardy-Hénon Equation with Regularly Varying Time Weights

We investigate the Cauchy problem for a semilinear parabolic equation driven by a mixed local-nonlocal diffusion operator of the form \[ \partial_t u - (Δ- (-Δ)^{\mathsf{s}})u = \mathsf{h}(t)|x|^{-b}|u|^p + t^\varrho \mathbf{w}(x), \qquad (x,t)\in \mathbb{R}^N\times (0,\infty), \] where $\mathsf{s}\in (0,1)$, $p>1$, $b\geq 0$, and $\varrho>-1$. The function $\mathsf{h}(t)$ is assumed to belong to the generalized class of regularly varying functions, while $\mathbf{w}$ is a prescribed spatial source. We first revisit the unforced case and establish sharp blow-up and global existence criteria in terms of the critical Fujita exponent, thereby extending earlier results to the wider class of time-dependent coefficients. For the forced problem, we derive nonexistence of global weak solutions under suitable growth conditions on $\mathsf{h}$ and integrability assumptions on $\mathbf{w}$. Furthermore, we provide sufficient smallness conditions on the initial data and the forcing term ensuring global-in-time mild solutions. Our analysis combines semigroup estimates for the mixed operator, test function methods, and asymptotic properties of regularly varying functions. To our knowledge, this is the first study addressing blow-up phenomena for nonlinear diffusion equations with such a class of time-dependent coefficients.

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arXiv2026

NLS equation with competing inhomogeneous nonlinearities: ground states, blow-up, and scattering

We investigate a class of nonlinear equations of Schrödinger type with competing inhomogeneous nonlinearities in the non-radial inter-critical regime, \begin{align*} i \partial_t u +Δu &=|x|^{-b_1} |u|^{p_1-2} u - |x|^{-b_2} |u|^{p_2-2}u \quad \mbox{in} \,\, \mathbb{R} \times \mathbb{R}^N, \end{align*} where $N \geq 1$, $b_1, b_2>0$ and $p_1,p_2>2$. First, we establish the existence/nonexistence, symmetry, decay, uniqueness, non-degeneracy and instability of ground states. Then, we prove the scattering versus blowup below the ground state energy threshold. Our approach relies on Tao's scattering criterion and Dodson-Murphy's Virial/Morawetz inequalities. We also obtain an upper bound of the blow-up rate. The novelty here is that the equation does not enjoy any scaling invariance due to the presence of competing nonlinearities and the singular weights prevent the invariance by translation in the space variable. To the best of authors knowledge, this is the first time when inhomegeneous NLS equation with a focusing leading order nonlinearity and a defocusing perturbation is investigated.

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arXiv2026

Forcing Effects on Finite-Time Blow-Up in Degenerate and Singular Parabolic Equations

We study the degenerate and singular parabolic equation with a forcing term \[ |x|^{σ_1}u_t = Δu + |x|^{σ_2}|u|^p + t^\varrho \mathbf{w}(x), \quad (t,x)\in(0,\infty)\times\mathbb{R}^N, \] where $N\ge 2$, $σ_1,σ_2>-2$, $\varrho>-1$, $p>1$, and $\mathbf{w}\in L^1(\mathbb{R}^N)$ is continuous. We establish critical exponents that sharply separate the regimes of global existence and finite-time blow-up. For $\varrho>0$, we prove that there is no weak global solution for all $p>1$. When $-1<\varrho<0$, we show that if \[ p < p^*:=\frac{N+σ_2-\varrho(2+σ_1)}{N-2-\varrho(2+σ_1)}, \] then every weak solution blows up in finite time, provided $\int\limits_{\mathbb{R}^N}\mathbf{w}(x)\,dx>0$. In the case $\varrho=0$, blow-up occurs for $p\le (N+σ_2)/(N-2)_+$ with $N\ge 2$. In contrast, for $p>p^*$ and under smallness conditions on the initial data and forcing term, we prove the existence of a unique global mild solution. The analysis relies on scaling transformations, semigroup estimates for degenerate operators, and a fixed-point argument in weighted-in-time Lebesgue spaces.

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arXiv2026

Normalized Standing Waves for the Focusing Inhomogeneous Schrödinger Equation with Spatially Growing Nonlinearity

We study the focusing inhomogeneous nonlinear Schrödinger equation $$ i\partial_t u + Δu = -|x|^b |u|^{p-1}u ,\quad (t,x)\in (0,\infty)\times\mathbb{R}^N, $$ with $b>0$ and $p>1$. Due to the spatial growth of the nonlinearity, standard compactness arguments do not apply and new difficulties arise. We first characterize ground state standing waves via a variational approach on the Nehari manifold and we establish some sharp stability and instability properties. In the $L^2$-subcritical regime, we prove the existence of normalized ground states by solving a constrained energy minimization problem in the radial energy space, and we show that the resulting set of minimizers is orbitally stable under the flow. In contrast, in the $L^2$-critical and supercritical regimes, ground state standing waves are shown to be strongly unstable by finite-time blow-up. Our results extend classical stability and instability theory for nonlinear Schrödinger equations to the case of spatially growing inhomogeneous nonlinearities.

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arXiv2026

On the Fujita Phenomenon for a Forced Spatio-Temporal Fractional Diffusion Equation

We investigate the Cauchy problem for a semilinear spatio--temporal fractional diffusion equation with a time-dependent forcing term: \[ \partial_t^αu + (-Δ)^{\mathsf{s}} u = |u|^p + t^σ\,\mathbf{w}(x), \quad (t,x) \in (0,\infty) \times \mathbb{R}^N, \] where $α,\mathsf{s}\in (0,1)$, $σ> -α$, and $\mathbf{w}$ is a given continuous function. Here $\partial_t^α$ denotes the Caputo fractional derivative. Our main results are threefold. First, we establish local-in-time existence of mild solutions and prove finite-time blow-up in the subcritical regime, under the positivity condition \[ \int\limits_{\mathbb{R}^N} \mathbf{w}(x)\,dx > 0. \] Second, in the supercritical case $-α< σ< 0$, we prove the global existence of solutions for sufficiently small initial data and forcing term, and we identify the corresponding critical exponent as \[ p_F=\frac{Nα-2\mathsf{s}σ}{Nα-2\mathsf{s}(α+σ)}. \] Finally, within this supercritical range, we obtain a more robust global existence result under weaker assumptions that require only local smallness and controlled growth of the data. To the best of our knowledge, a sharp Fujita-type threshold for fully spatio-temporal fractional diffusion equations with time-growing external forcing has not been previously established.

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arXiv2025

Long-Time Asymptotics for Subordinated Fractional Diffusion Equations

We study the long-time behavior of solutions to a class of evolution equations arising from random-time changes driven by subordinators. Our focus is on fractional diffusion equations involving mixed local and nonlocal operators. By combining techniques from probability theory, asymptotic analysis, and partial differential equations (PDEs), we characterize the dynamics of the subordinated solutions. This approach extends classical fractional dynamics and establishes a deeper connection between stochastic processes and deterministic PDEs.

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arXiv2025

Damping Effects on Global Existence and Scattering for an Inhomogeneous NLS Equation with Inverse-Square Potential

This work explores the global existence and scattering behavior of solutions to a damped, inhomogeneous nonlinear Schrodinger equation featuring a time-dependent damping term, an inverse-square potential, and an inhomogeneous nonlinearity. We establish global well-posedness in the energy space for subcritical, mass-critical, and energy-critical regimes, using Strichartz estimates, Hardy inequalities, and Gagliardo-Nirenberg-type estimates. For sufficiently large damping, we highlight how the interplay between damping, singular potentials, and inhomogeneity influences the dynamics. Our results extend existing studies and offer new insights into the long-time behavior of solutions in this more general setting. To the best of our knowledge, this is the first study to address the combined effects of inverse-square potential, inhomogeneous (or even homogeneous) nonlinearity, and damping in the context of the NLS equation.

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arXiv2025

The lifespan of solutions of semilinear wave equation with weighted nonlinearity

We investigate the lifespan of solutions to a specific variant of the semilinear wave equation, which incorporates weighted nonlinearity $$ u_{tt}-u_{xx} =|x|^α|u|^p, \quad\mbox{for}\;\;\; (t,x)\in (0,\infty)\times\mathbb{R}, $$ where $p>1$, $α\in\mathbb{R}$. We explore the behavior of solutions for small initial data, considering the influence of weighted nonlinearities on the lifespan.

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arXiv2025

Well-posedness and linearization for a semilinear wave equation with spatially growing nonlinearity

We study the initial value problem for a defocusing semi-linear wave equation with spatially growing nonlinearity. By employing Moser-Trudinger type inequalities and Strichartz estimates, we establish global well-posedness in the energy space for radially symmetric initial data. Additionally, we derive the linearization of energy-bounded solutions. The main challenge in our analysis arises from the spatial growth of the nonlinearity at infinity, which prevents the direct application of Sobolev embeddings or Hardy inequalities to control the potential energy. The main novelty in this work lies in overcoming this challenge within the radial framework through the combined application of the Strauss inequality and Strichartz estimates.

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arXiv2025

On the Nonexistence of Global Solutions for Nonlocal Parabolic Equations with Forcing Terms

The purpose of this work is to analyze the well-posedness and blow-up behavior of solutions to the nonlocal semilinear parabolic equation with a forcing term: \[ \partial_t u - Δu = \|u(t)\|_{q}^α|u|^p + t^{\varrho} \mathbf{w}(x) \quad \text{in} \quad \mathbb{R}^N \times (0, \infty), \] where $N \geq 1$, $p, q \geq 1$, $α\geq 0$, $\varrho > -1$, and $\mathbf{w}(x)$ is a suitably given continuous function. The novelty of this work, compared to previous studies, lies in considering a nonlocal nonlinearity $\|u(t)\|_{q}^α|u|^p$ and a forcing term $t^{\varrho} \mathbf{w}(x)$ that depend on both time and space variables. This combination introduces new challenges in understanding the interplay between the nonlocal structure of the equation and the spatio-temporal forcing term. Under appropriate assumptions, we establish the global existence of solutions for small initial data in Lebesgue spaces when the exponent $p$ exceeds a critical value. In contrast, we show that the global existence cannot hold for $p$ below this critical value, provided the additional condition $\int_{\mathbb{R}^N} \mathbf{w}(x) \, dx > 0$ is satisfied. The main challenge in this analysis lies in managing the complex interaction between the nonlocal nonlinearity and the forcing term, which significantly influences the behavior of solutions.

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