arXiv subjects

Mohamed Ali Hamza

Publications and source records attributed to Mohamed Ali Hamza.

At least 19 recordsLinked to original sources

arXiv2026

Blow-up criteria and lifespan estimates for semilinear wave equations with time-dependent damping and mass

We consider semilinear wave equations with time-dependent damping and mass and a derivative-type nonlinearity. By applying a Liouville transformation and solving a Volterra integral equation posed from infinity, we construct a positive exact solution to the adjoint equation without using an explicit representation of the linear propagator. This solution is used to derive a blow-up criterion for energy solutions with finite propagation, together with an upper bound for the lifespan. If the primitive of the damping coefficient grows at most logarithmically, we obtain the full shifted Glassey range, including the critical exponent, and the corresponding polynomial and exponential lifespan estimates. The result applies independently of the sign of the scale-invariant discriminant and therefore includes the mass-dominant regime. It also covers non-integrable oscillatory perturbations of the damping coefficient when their contributions are canceled by the corresponding mass terms.

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arXiv2026

Asymptotic profiles for the Cauchy problem of semilinear beam equation with two variable coefficients in the subcritical case

In this article, we investigate the asymptotic profile of solutions to the Cauchy problem for a nonlinear beam equation with two variable coefficients in the subcritical nonlinear case. In contrast to our previous result [6], in which the asymptotic profile is governed by the linear heat kernel and the nonlinear effect is asymptotically negligible, the asymptotic profile in the present setting is described by a self-similar solution to the associated nonlinear parabolic equation (constructed in Brezis-Peletier-Terman [1]). The proof relies on delicate energy estimates in weighted spaces formulated in parabolic self-similar variables.

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arXiv2026

On the blow-up of solutions to scale-invariant wave equations with damping and mass: Beyond the positive discriminant restriction

This paper investigates the blow-up of solutions to scale-invariant semilinear wave equations featuring the damping term $\fracμ{1+t} \partial_t u$, the mass term $\frac{ν^2}{(1+t)^2} u$, and a time-derivative nonlinearity $| \partial_t u |^p$. The principal contribution of this work is the demonstration that the sign of the discriminant $δ= (μ-1)^2 - 4ν^2$ is not a structural prerequisite for determining the blow-up range. Indeed, we show that even in the regime $δ< 0$, the blow-up region remains invariant and is uniquely determined by the shifted dimension $n+μ$, aligning with the Glassey-type critical exponent. Our result suggest that the classical restriction $δ\ge 0$ is due to a technical tool rather than an intrinsic feature of the blow-up mechanism.

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arXiv2026

The blow-up rate for a log non-scaling invariant semilinear wave equation in the conformal regime

We consider the blow-up behavior of solutions to the semilinear wave equation $$ \partial_t^2 u - Δu = |u|^{p-1}u \ln^a(u^2+2), \ (x,t)\in \mathbb{R}^n \times [0,T),$$ in the conformal case $ p = p_c = 1 + \frac{4}{n-1}$. Previous results in \cite{HZjmaa2020, HZ2022} show that for $ a \in \mathbb{R} $, solutions in the subconformal regime $ p < p_c $ blow up with a Type~I rate at any non-characteristic point. The objective of this work is to extend this blow-up rate to the conformal regime under the assumption $a<0$. We establish an a priori upper bound for any blow-up solution and construct a Lyapunov functional in similarity variables. The resulting functional exhibits only weak dissipation, which necessitates delicate energy arguments to obtain the sharp blow-up rate in the conformal case. To the best of our knowledge, this provides the first result for the blow-up rate in a critical framework for an evolution problem where the scaling symmetry is broken.

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arXiv2025

Critical exponent for the one-dimensional wave equation with a space-dependent scale invariant damping and time derivative nonlinearity

We investigate in this paper the Cauchy problem of the one-dimensional wave equation with space-dependent damping of the form $μ_0(1+x^2)^{-1/2}$, where $μ_0>0$, and time derivative nonlinearity. We establish global existence of mild solutions for small data compactly supported by employing energy estimates within suitable Sobolev spaces of the associated homogeneous problem. Furthermore, we derive a blow-up result under some positive initial data by employing the test function method. This shows that the critical exponent is given by $p_G(1+μ_0)=1+2/μ_0$, when $μ_0\in (0,1]$, where $p_G$ is the Glassey exponent. To the best of our knowledge, this constitutes the first identification of the critical exponent range for this class of equations. As by product, we extend the global existence result to a more general class of space/time nonlinearities of the form $f(\partial_tu,\partial_x u)=|\partial_x u|^{q}$ or $f(\partial_tu,\partial_x u)=|\partial_tu|^{p}|\partial_x u|^{q}$, with $p,q>1$.

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arXiv2025

Global existence for wave equations with scale-invariant time-dependent damping and time derivative nonlinearity

This paper addresses the Cauchy problem for wave equations with scale-invariant time-dependent damping and nonlinear time-derivative terms, modeled as $$\partial_{t}^2u- Δu +\fracμ{1+t}\partial_tu= f(\partial_tu), \quad x\in \mathbb{R}^n, t>0,$$ where $f(\partial_tu)=|\partial_tu|^p $ or $|\partial_tu|^{p-1}\partial_tu$ with $p>1$ and $μ>0$. We prove global existence of small data solutions in low dimensions $1\leq n\leq 3$ by using energy estimates in appropriate Sobolev spaces. Our primary contribution is an existence result for $p>1+\frac2μ$, in the one-dimensional case, when $μ\le 2$, which in conjunction with prior blow-up results from \cite{Our2}, establish that the critical exponent for small data solutions in one dimension is $p_G(1,μ)=1+\frac2μ$, when $μ\le 2$. To the best of our knowledge, this is the first identification of the critical exponent range for the time-dependent damped wave equations with scale-invariant and time-derivative nonlinearity.

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arXiv2025

A better bound on blow-up rate for the superconformal semilinear wave equation

We consider the semilinear wave equation in higher dimensions with superconformal power nonlinearity. The purpose of this paper is to give a new upper bound on the blow-up rate in some space-time integral, showing a $|\log(T-t)|^q$ improvement in comparison with previous results obtained in \cite{HZdcds13,KSVsurc12}.

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arXiv2024

Blow-up result for a weakly coupled system of two Euler-Poisson-Darboux-Tricomi equations with time derivative nonlinearity

We study in this article the blow-up of solutions to a coupled semilinear wave equations which are characterized by linear damping terms in the \textit{scale-invariant regime}, time-derivative nonlinearities, mass terms and Tricomi terms. The latter are specifically of great interest from both physical and mathematical points of view since they allow the speeds of propagation to be time-dependent ones. However, we assume in this work that both waves are propagating with the same speeds. Employing this fact together with other hypotheses on the aforementioned parameters (mass and damping coefficients), we obtain a new blow-up region for the system under consideration, and we show a lifespan estimate of the maximal existence time.

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arXiv2023

Blow-up and lifespan estimate for the generalized tricomi equation with the scale-invariant damping and time derivative nonlinearity on exterior domain

The article is devoted to investigating the initial boundary value problem for the damped wave equation in the scale-invariant case with time-dependent speed of propagation on the exterior domain. By presenting suitable multipliers and applying the test-function technique, we study the blow-up and the lifespan of the solutions to the problem with derivative-type nonlinearity $ \d u_{tt}-t^{2m}Δu+\fracμ{t}u_t=|u_t|^p, \quad \mbox{in}\ Ω^{c}\times[1,\infty),$ that we associate with appropriate small initial data.

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arXiv2023

Asymptotic profiles for the Cauchy problem of damped beam equation with two variable coefficients and derivative nonlinearity

In this article we investigate the asymptotic profile of solutions for the Cauchy problem of the nonlinear damped beam equation with two variable coefficients: \[ \partial_t^2 u + b(t) \partial_t u - a(t) \partial_x^2 u + \partial_x^4 u = \partial_x \left( N(\partial_x u) \right). \] In the authors' previous article [17], the asymptotic profile of solutions for linearized problem ($N \equiv 0$) was classified depending on the assumptions for the coefficients $a(t)$ and $b(t)$ and proved the asymptotic behavior in effective damping cases. We here give the conditions of the coefficients and the nonlinear term in order that the solution behaves as the solution for the heat equation: $b(t) \partial_t u - a(t) \partial_x^2 u=0$ asymptotically as $t \to \infty$.

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arXiv2022

Blow-up and lifespan estimate for wave equations with critical damping term of space-dependent type related to Glassey conjecture

The main purpose of the present paper is to study the blow-up problem of the wave equation with space-dependent damping in the \textit{scale-invariant case} and time derivative nonlinearity with small initial data. Under appropriate initial data which are compactly supported, by using a test function method and taking into account the effect of the damping term ($\fracμ{\sqrt{1+|x|^2}}u_t$), we provide that in higher dimensions the blow-up region is given by $p \in (1, p_G(N+μ)]$ where $p_G(N)$ is the Glassey exponent. Furthermore, we shall establish a blow-up region, independent of $μ$ given by $p\in (1, 1+\frac{2}{N}),$ for appropriate initial data in the energy space with noncompact support.

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arXiv2022

Improvement on the blow-up for a weakly coupled wave equations with scale-invariant damping and mass and time derivative nonlinearity

An improvement of [18] on the blow-up region and the lifespan estimate of a weakly coupled system of wave equations with damping and mass in the scale-invariant case and with time-derivative nonlinearity is obtained in this article. Indeed, thanks to a better understanding of the dynamics of the solutions, we give here a better characterization of the blow-up region. Furthermore, the techniques used in this article may be extended to other systems and interestingly they simplify the proof of the blow-up result in [3] which is concerned with the single wave equation in the same context as in the present work.

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arXiv2022

Blow-up and lifespan estimates for a damped wave equation in the Einstein-de Sitter spacetime with nonlinearity of derivative type

In this article, we investigate the blow-up for local solutions to a semilinear wave equation in the generalized Einstein - de Sitter spacetime with nonlinearity of derivative type. More precisely, we consider a semilinear damped wave equation with a time-dependent and not summable speed of propagation and with a time-dependent coefficient for the linear damping term with critical decay rate. We prove in this work that the results obtained in a previous work, where the damping coefficient takes two particular values $0$ or $2$, can be extended for any positive damping coefficient. In the blow-up case, the upper bound of the exponent of the nonlinear term is given, and the lifespan estimate of the global existence time is derived as well.

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arXiv2022

The blow-up rate for a non-scaling invariant semilinear heat equation

We consider the semilinear heat equation $$\partial_t u -Δu =f(u), \quad (x,t)\in \mathbb{R}^N\times [0,T),\qquad (1)$$ with $f(u)=|u|^{p-1}u\log^a (2+u^2)$, where $p>1$ is Sobolev subcritical and $a\in \mathbb{R}$. We first show an upper bound for any blow-up solution of (1). Then, using this estimate and the logarithmic property, we prove that the exact blow-up rate of any singular solution of (1) is given by the ODE solution associated with (1), namely $u' =|u|^{p-1}u\log^a (2+u^2)$. In other terms, all blow-up solutions in the Sobolev subcritical range are Type I solutions. Up to our knowledge, this is the first determination of the blow-up rate for a semilinear heat equation where the main nonlinear term is not homogeneous.

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arXiv2021

Nonexistence result for the generalized Tricomi equation with the scale-invariant damping, mass term and time derivative nonlinearity

In this article, we consider the damped wave equation in the \textit{scale-invariant case} with time-dependent speed of propagation, mass term and time derivative nonlinearity. More precisely, we study the blow-up of the solutions to the following equation: $$ (E) \quad u_{tt}-t^{2m}Δu+\fracμ{t}u_t+\frac{ν^2}{t^2}u=|u_t|^p, \quad \mbox{in}\ \mathbb{R}^N\times[1,\infty), $$ that we associate with small initial data. Assuming some assumptions on the mass and damping coefficients, $ν$ and $μ>0$, respectively, that the blow-up region and the lifespan bound of the solution of $(E)$ remain the same as the ones obtained for the case without mass, {\it i.e.} $ν=0$ in $(E)$. The latter case constitutes, in fact, a shift of the dimension $N$ by $\fracμ{1+m}$ compared to the problem without damping and mass. Finally, we think that the new bound for $p$ is a serious candidate to the critical exponent which characterizes the threshold between the blow-up and the global existence regions.

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arXiv2021

A note on the nonexistence of global solutions to the semilinear wave equation with nonlinearity of derivative-type in the generalized Einstein-de Sitter spacetime

In this paper, we establish blow-up results for the semilinear wave equation in generalized Einstein-de Sitter spacetime with nonlinearity of derivative type. Our approach is based on the integral representation formula for the solution to the corresponding linear problem in the one-dimensional case, that we will determine through Yagdjian's Integral Transform approach. As upper bound for the exponent of the nonlinear term, we discover a Glassey-type exponent which depends both on the space dimension and on the Lorentzian metric in the generalized Einstein-de Sitter spacetime.

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arXiv2021

The blow-up rate for a non-scaling invariant semilinear wave equations in higher dimensions

We consider the semilinear wave equation $$\partial_t^2 u -Δu =f(u), \quad (x,t)\in \mathbb R^N\times [0,T),\qquad (1)$$ with $f(u)=|u|^{p-1}u\log^a (2+u^2)$, where $p>1$ and $a\in \mathbb R$, with subconformal power nonlinearity. We will show that the blow-up rate of any singular solution of (1) is given by the ODE solution associated with $(1)$, The result in one space dimension, has been proved in \cite{HZjmaa2020}. Our goal here is to extend this result to higher dimensions.

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arXiv2020

Blow-up and lifespan estimate for the generalized Tricomi equation with mixed nonlinearities

We study in this article the blow-up of the solution of the generalized Tricomi equation in the presence of two mixed nonlinearities, namely we consider $$ (Tr) \hspace{1cm} u_{tt}-t^{2m}Δu=|u_t|^p+|u|^q, \quad \mbox{in}\ \mathbb{R}^N\times[0,\infty),$$ with small initial data, where $m\ge0$.\\ For the problem $(Tr)$ with $m=0$, which corresponds to the uniform wave speed of propagation, it is known that the presence of mixed nonlinearities generates a new blow-up region in comparison with the case of a one nonlinearity ($|u_t|^p$ or $|u|^q$). We show in the present work that the competition between the two nonlinearities still yields a new blow region for the Tricomi equation $(Tr)$ with $m\ge0$, and we derive an estimate of the lifespan in terms of the Tricomi parameter $m$. As an application of the method developed for the study of the equation $(Tr)$ we obtain with a different approach the same blow-up result as in \cite{Lai2020} when we consider only one time-derivative nonlinearity, namely we keep only $|u_t|^p$ in the right-hand side of $(Tr)$.

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