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Mohamed Majdoub

Publications and source records attributed to Mohamed Majdoub.

At least 37 records · Page 2Linked to original sources
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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arXiv2025

Heat equations associated to harmonic oscillator with exponential nonlinearity

We investigate the Cauchy problem for a heat equation involving a fractional harmonic oscillator and an exponential nonlinearity. We establish local well-posedness within the appropriate Orlicz spaces. Through the examination of small initial data in suitable Orlicz spaces, we obtain the existence of global weak-mild solutions. Additionally, precise decay estimates are presented for large time, indicating that the decay rate is influenced by the nonlinearity's behavior near the origin. Moreover, we highlight that the existence of local nonnegative classical solutions is no longer guaranteed when certain nonnegative initial data is considered within the appropriate Orlicz space.

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arXiv2025

Energy scattering for the unsteady damped nonlinear Schrodinger equation

We investigate the large time behavior of the solutions to the nonlinear focusing Schrödinger equation with a time-dependent damping in the energy sub-critical regime. Under non classical assumptions on the unsteady damping term, we prove some scattering results in the energy space.

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arXiv2025

Radial 3D Focusing Energy Critical INLS equations with defocusing perturbation: Ground states, Scattering, and Blow-up

We investigate the following inhomogeneous nonlinear Schrödinger equation in the radial regime, featuring a focusing energy-critical nonlinearity and a defocusing perturbation: $$ i\partial_t u +Δu =|x|^{-a} |u|^{p-2} u - |x|^{-b} |u|^{4-2b}u \quad \mbox{in} \,\, \mathbb{R}_t \times \mathbb{R}_x^3, $$ where $0<a$, $b<2$ and $2+\frac{4-2a}{3}< p\leq 6-2a$. First, we establish the existence and nonexistence of ground states, along with their quantitative properties. Subsequently, we analyze the dichotomy between scattering and blow-up for solutions with energy below the ground-state energy threshold. An intriguing feature of this equation is the lack of scaling invariance, which arises from the competing effects of the inhomogeneous nonlinearities. Additionally, the presence of singular weights breaks translation invariance in the spatial variable, introducing further complexity to the analysis. To the best of our knowledge, this work represents the first comprehensive study of the inhomogeneous nonlinear Schrödinger equation with a leading-order focusing energy-critical inhomogeneous nonlinearity and a defocusing perturbation. Our results provide new insights into the interplay between these competing nonlinearities and their influence on the dynamics of solutions.

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arXiv2023

Existence and nonexistence of global solutions for time-dependent damped NLS equations

We investigate the Cauchy problem for the nonlinear Schrödinger equation with a time-dependent linear damping term. Under non standard assumptions on the loss dissipation, we prove the blow-up in the inter-critical regime, and the global existence in the energy subcritical case. Our results generalize and improve the ones in [9, 11, 21].

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arXiv2023

Well-posedness and scattering for a 2D inhomogeneous NLS with Aharonov-Bohm magnetic potential

We consider the magnetic nonlinear inhomogeneous Schrödinger equation $$i\partial_t u -\left(-i\nabla+\fracα{|x|^2}(-x_2,x_1)\right)^2 u =\pm|x|^{-\varrho}|u|^{p-1}u,\quad (t,x)\in \mathbb{R}\times \mathbb{R}^2,$$ where $α\in\mathbb{R}\setminus\mathbb{Z},\,\varrho>0,\,p>1$. We prove a dichotomy of global existence and scattering versus blow-up of energy solutions under the ground state threshold in the inter-critical regime. The scattering is obtained by using the new approach of Dodson-Murphy (A new proof of scattering below the ground state for the 3D radial focusing cubic NLS, {Proc. Am. Math. Soc.} (2017)). This method is based on Tao's scattering criteria and Morawetz estimates. The novelty here is twice: we investigate the case $\varrhoα\neq0$ and we consider general energy initial data (not necessarily radially symmetric).

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arXiv2022

On the Fujita exponent for a Hardy-Hénon equation with a spatial-temporal forcing term

The purpose of this work is to analyze the wellposedness and the blow-up of solutions of the higher-order parabolic semilinear equation \[ u_t+(-Δ)^{d}u=|x|^α|u|^{p}+ζ(t){\mathbf w}(x) \ \quad\mbox{for } (x,t)\in\mathbb{R}^{N}\times(0,\infty), \] where $d\in (0,1)\cup \mathbb{N}$, $p>1$, $-α\in(0,\min(2d,N))$ or $α\geq 0$ and $ζ$ as well as ${\mathbf w}$ are suitable given functions. Given $p\geq \frac{N-2dσ+α}{N-2dσ-2d}$ and setting $p_c=\frac{N(p-1)}{2d+α}$, $\ell=\frac{N p_c}{N+2(σ+1)d p_c}$, we prove that for any data $u_0\in L^{p_c,\infty}(\mathbb{R}^N)$ and $\textbf{w}\in L^{\ell,\infty}(\mathbb{R}^N)$ with small norms there exists a unique global-in-time solution under the hypotheses $ζ(t)=t^σ$, $σ\in (-1,0)$ and $N>2d$ in the space $C_{b}([0,\infty);L^{p_c,\infty}(\mathbb{R}^N))$. As a by-product, small Lebesgue data global existence follows and in particular, unconditional uniqueness holds in $C_{b}([0,\infty);L^{p_c}(\mathbb{R}^N))$ provided $p\in (\frac{N+α}{N-2d},\infty)$. If either $m\in (-\infty,0]$ and $p\in (1,\frac{N-2dm+α}{N-2dm-2d})$ or $m>0$ and $p>1$ where $ζ(t)=O(t^m)$, $t\rightarrow\infty$ ($m\in \mathbb{R}$), then all solutions blow up under the additional condition $\int_{\mathbb{R}^N}\textbf{w}(x)\,dx>0$. As a consequence, we deduce that the corresponding Fujita critical exponent is a function of $σ$ and reads $p_{F}(σ)=\frac{N-2dσ+α}{N-2dσ-2d}$ if $-1<σ<0$ and infinity otherwise.

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arXiv2022

Long time dynamics for the focusing inhomogeneous fractional Schrödinger equation

We consider the following fractional NLS with focusing inhomogeneous power-type nonlinearity $$i\partial_t u -(-Δ)^s u + |x|^{-b}|u|^{p-1}u=0,\quad (t,x)\in \mathbb{R}\times \mathbb{R}^N,$$ where $N\geq 2$, $1/2<s<1$, $0<b<2s$ and $1+\frac{2(2s-b)}{N}<p<1+\frac{2(2s-b)}{N-2s}$. We prove the ground state threshold of global existence and scattering versus finite time blow-up of energy solutions in the inter-critical regime with spherically symmetric initial data. The scattering is proved by the new approach of Dodson-Murphy ({Proc. Am. Math. Soc.} {145}: {4859--4867}, 2017). This method is based on Tao's scattering criteria and Morawetz estimates. One describes the threshold using some non-conserved quantities in the spirit of the recent paper by Dinh (Discr. Cont. Dyn. Syst. 40: 6441--6471, 2020). The radial assumption avoids a loss of regularity in Strichartz estimates. The challenge here is to overcome two main difficulties. The first one is the presence of the non-local fractional Laplacian operator. The second one is the presence of a singular weight in the non-linearity. The greater part of this paper is devoted to prove the scattering of global solutions in $H^s(\mathbb{R}^N)$.

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arXiv2022

Global existence and asymptotic behavior for a reaction-diffusion system with unbounded coefficients

We consider a reaction-diffusion system which may serve as a model for a ferment catalytic reaction in chemistry. The model consists of a system of reaction diffusion equations with unbounded time dependent coefficients and different polynomial reaction terms. An exponential decay of the globally bounded solutions is proved. The key tool of the proofs are properties of analytic semigroups and some inequalities.

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arXiv2022

Existence and regularity of source-type self-similar solutions for stable thin-film equations

We investigate the existence and the boundary regularity of source-type self-similar solutions to the thin-film equation $h_t=-(h^nh_{zzz})_z+(h^{n+3})_{zz},$ $ t>0,\; z\in \mathbb{R};\; h(0,z)= ωδ(z)$ where $n\in (\frac{3}{2},3),\; ω> 0$ and $δ$ is the Dirac mass at the origin. It is known that the leading order expansion near the edge of the support coincides with that of a traveling-wave solution for the standard thin-film equation: $h_t=-(h^nh_{zzz})_z$. In this paper we sharpen this result, proving that the higher order corrections are analytic with respect to three variables: the first one is just the {spatial} variable, whereas the second and the third (except for $n = 2$) are irrational powers of it. It is known that this third variable does not appear for the thin-film equation without gravity.

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arXiv2021

Long time dynamics and blow-up for the focusing inhomogeneous nonlinear Schrödinger equation with spatial growing nonlinearity

We investigate the Cauchy problem for the focusing inhomogeneous nonlinear Schrödinger equation $i \partial_t u + Δu = - |x|^b |u|^{p-1} u$ in the radial Sobolev space $H^1_{\text{rad}}(\mathbb{R}^N)$, where $b>0$ and $p>1$. We show the global existence and energy scattering in the inter-critical regime, i.e., $p>\frac{N+4+2b}{N}$ and $p<\frac{N+2+2b}{N-2}$ if $N\geq 3$. We also obtain blowing-up solutions for the mass-critical and mass-supercritical nonlinearities. The main difficulty, coming from the spatial growing nonlinearity, is overcome by refined Gagliardo-Nirenberg type inequalities. Our proofs are based on improved Gagliardo-Nirenberg inequalities, the Morawetz-Sobolev approach of Dodson and Murphy, radial Sobolev embeddings, and localized virial estimates.

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arXiv2021

Well-posedness and blow-up for an inhomogeneous semilinear parabolic equation

We consider the large-time behavior of sign-changing solutions of the inhomogeneous equation $u_t-Δu=|x|^α|u|^{p}+ζ(t)\,{\mathbf w}(x)$ in $(0,\infty)\times\mathbb{R}^N$, where $N\geq 3$, $p>1$, $α>-2$, $\z, {\mathbf w}$ are continuous functions such that $ζ(t)=t^σ$ or $ζ(t)\sim t^σ$ as $t\to 0$, $ζ(t)\sim t^m$ as $t\to\infty$ . We obtain local existence for $σ>-1$. We also show the following: \begin{itemize} \item If $m\leq 0$, $p<\frac{N-2m+α}{N-2m-2}$ and $\int_{\mathbb{R}^N}{\mathbf w}(x)dx>0$, then all solutions blow up in finite time; \item If $m> 0$, $p>1$ and $\int_{\mathbb{R}^N}{\mathbf w}(x)dx>0$, then all solutions blow up in finite time; \item If $ζ(t)=t^σ$ with $-1<σ<0$, then for $u_0:=u(t=0)$ and ${\mathbf w}$ sufficiently small the solution exists globally. \end{itemize} We also discuss lower dimensions. The main novelty in this paper is that blow up depends on the behavior of $ζ$ at infinity.

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arXiv2020

Long time dynamics for the focusing nonlinear Schrödinger equation with exponential nonlinearities

In this paper, we study the focusing nonlinear Schrödinger equation with exponential nonlinearities \[ i \partial_t u + Δu = - \left(e^{4π|u|^2} - 1 - 4πμ|u|^2 \right) u, \quad u(0) = u_0 \in H^1, \quad (t,x) \in \mathbb{R} \times \mathbb{R}^2, \] where $μ\in \{0, 1\}$. By using variational arguments, we first derive invariant sets where the global existence and finite time blow-up occur. In particular, we obtain sharp thresholds for global existence and finite time blow-up. In the case $μ=1$, by adapting a recent argument of Arora-Dodson-Murphy \cite{ADM}, we study the long time dynamics of global solutions. It turns out that either there exist $t_n\rightarrow +\infty$ and $R_n \rightarrow \infty$ such that $u(t_n)$ vanishes inside $B(0,R_n)$ for all $n\geq 1$ or the solution scatters in $H^1$.

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arXiv2020

Relaxation to equilibrium in the one-dimensional thin-film equation with partial wetting

We investigate the large time behavior of compactly supported solutions for a one-dimensional thin-film equation with linear mobility in the regime of partial wetting. We show the stability of steady state solutions. The proof uses the Lagrangian coordinates. Our method is to establish and exploit differential relations between the energy and the dissipation as well as some interpolation inequalities. Our result is different from earlier results because here we consider solutions with finite mass.

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