arXiv subjects
Publications and source records attributed to Mohamed Majdoub.
We study a class of initial boundary value problems of hyperbolic type. A new topological approach is applied to prove the existence of non-negative classical solutions. The arguments are based upon a recent theoretical result.
In this paper we consider the initial value {problem $\partial_{t} u- Δu=f(u),$ $u(0)=u_0\in exp\,L^p(\mathbb{R}^N),$} where $p>1$ and $f : \mathbb{R}\to\mathbb{R}$ having an exponential growth at infinity with $f(0)=0.$ Under smallness condition on the initial data and for nonlinearity $f$ {such that $|f(u)|\sim \mbox{e}^{|u|^q}$ as $|u|\to \infty$,} $|f(u)|\sim |u|^{m}$ as $u\to 0,$ $0 1$, we show that the solution is global. Moreover, we obtain decay estimates in Lebesgue spaces for large time which depend on $m.$
This paper deals with the 2-D Schrödinger equation with time-oscillating exponential nonlinearity $i\partial_t u+Δu= θ(ωt)\big(e^{4π|u|^2}-1\big)$, where $θ$ is a periodic $C^1$-function. We prove that for a class of initial data $u_0 \in H^1(\mathbb{R}^2)$, the solution $u_ω$ converges, as $|ω|$ tends to infinity to the solution $U$ of the limiting equation $i\partial_t U+ΔU= I(θ)\big(e^{4π|U|^2}-1\big)$ with the same initial data, where $I(θ)$ is the average of $θ$.
We investigate the defocusing inhomogeneous nonlinear Schrödinger equation $$ i \partial_tu + Δu = |x|^{-b} \left({\rm e}^{α|u|^2} - 1- α|u|^2 \right) u, \quad u(0)=u_0, \quad x \in \mathbb{R}^2, $$ with $0<b<1$ and $α=2π(2-b)$. First we show the decay of global solutions by assuming that the initial data $u_0$ belongs to the weighted space $Σ(\mathbb{R}^2)=\{\,u\in H^1(\mathbb{R}^2) \ : \ |x|u\in L^2(\mathbb{R}^2)\,\}$. Then we combine the local theory with the decay estimate to obtain scattering in $Σ$ when the Hamiltonian is below the value $\frac{2}{(1+b)(2-b)}$.
In this paper we consider the problem: $\partial_{t} u- Δu=f(u),\; u(0)=u_0\in \exp L^p(\R^N),$ where $p>1$ and $f : \R\to\R$ having an exponential growth at infinity with $f(0)=0.$ We prove local well-posedness in $\exp L^p_0(\R^N)$ for $f(u)\sim \mbox{e}^{|u|^q},\;0 0,$ $\displaystyle\liminf_{s\to \infty}\left(f(s)\,{\rm{e}}^{-λs^p}\right)>0,$ then non-existence occurs in $\exp L^p(\R^N).$ Under smallness condition on the initial data and for exponential nonlinearity $f$ such that $|f(u)|\sim |u|^{m}$ as $u\to 0,$ ${N(m-1)\over 2}\geq p$, we show that the solution is global. In particular, $p-1>0$ sufficiently small is allowed. Moreover, we obtain decay estimates in Lebesgue spaces for large time which depend on $m$.
We investigate the initial value problem for a defocusing nonlinear Schrödinger equation with weighted exponential nonlinearity $$ i\partial_t u+Δu=\frac{u}{|x|^b}(e^{α|u|^2}-1); \qquad (t,x) \in \mathbb{R}\times\mathbb{R}^2, $$ where $0< b <1$ and $α=2π(2-b)$. We establish local and global well-posedness in the subcritical and critical regimes.
In this paper we prove local well-posedness in Orlicz spaces for the biharmonic heat equation $\partial_{t} u+ Δ^2 u=f(u),\;t>0,\;x\in\R^N,$ with $f(u)\sim \mbox{e}^{u^2}$ for large $u.$ Under smallness condition on the initial data and for exponential nonlinearity $f$ such that $f(u)\sim u^m$ as $u\to 0,$ $m$ integer and $N(m-1)/4\geq 2$, we show that the solution is global. Moreover, we obtain a decay estimates for large time for the nonlinear biharmonic heat equation as well as for the nonlinear heat equation. Our results extend to the nonlinear polyharmonic heat equation.
We prove global existence of strong solutions to the drift-diffusion-Maxwell system in two space dimension. We also provide an exponential growth estimate for the $H^1$ norm of the solution.
In this paper we study the equations governing the unsteady motion of an incompressible homogeneous generalized second grade fluid subject to periodic boundary conditions. We establish the existence of global-in-time strong solutions for shear thickening flows in the two and three dimensional case. We also prove uniqueness of such solution without any smallness condition on the initial data or restriction on the material moduli.
In a recent paper, Struwe considered the Cauchy problem for a class of nonlinear wave and Scrödinger equations. Under some assumptions on the nonlinearities, it was shown that uniqueness of classical solutions can be obtained in the much larger class of distribution solutions satisfying the energy inequality. As pointed out in [38], the conditions on the nonlinearities are satisfied for any polynomial growth but they fail to hold for higher growth (for example ${\rm e}^{u^2}$). Our aim here is to improve Struwe's result by showing that uniqueness holds for more general nonlinearities including higher growth or oscillations.
We prove a Log Log inequality with a sharp constant in four dimensions for radially symmetric functions. We also show that the constant in the Log estimate is almost sharp.
We investigate an evolutive system of non-linear partial differential equations derived from Oldroyd models on Non-Newtonian flows. We prove global existence of weak solutions, in the case of a smooth bounded domain, for general initial data. The results hold true for the periodic case.
This paper is devoted to the description of the lack of compactness of the Sobolev embedding of $H^1(\R^2)$ in the critical Orlicz space ${\cL}(\R^2)$. It turns out that up to cores our result is expressed in terms of the concentration-type examples derived by J. Moser in \cite{M} as in the radial setting investigated in \cite{BMM}. However, the analysis we used in this work is strikingly different from the one conducted in the radial case which is based on an $L^ \infty$ estimate far away from the origin and which is no longer valid in the general framework. Within the general framework of $H^1(\R^2)$, the strategy we adopted to build the profile decomposition in terms of examples by Moser concentrated around cores is based on capacity arguments and relies on an extraction process of mass concentrations. The essential ingredient to extract cores consists in proving by contradiction that if the mass responsible for the lack of compactness of the Sobolev embedding in the Orlicz space is scattered, then the energy used would exceed that of the starting sequence.
We investigate the initial value problem for a semilinear heat equation with exponential-growth nonlinearity in two space dimension. First, we prove the local existence and unconditional uniqueness of solutions in the Sobolev space $H^1(\R^2)$. The uniqueness part is non trivial although it follows Brezis-Cazenave's proof \cite{Br} in the case of monomial nonlinearity in dimension $d\geq3$. Next, %Following Caffarelli-Vasseur \cite{cv}, we show that in the defocusing case our solution is bounded, and therefore exists for all time. In the focusing case, we prove that any solution with negative energy blows up in finite time. Lastly, we show that the unconditional result is lost once we slightly enlarge the Sobolev space $H^1(\R^2)$. The proof consists in constructing a singular stationnary solution that will gain some regularity when it serves as initial data in the heat equation. The Orlicz space appears to be appropriate for this result since, in this case, the potential term is only an integrable function.
In order to describe behavior of various liquid-like materials at high pressures, incompressible fluid models with pressure dependent viscosity seem to be a suitable choice. In the context of implicit constitutive relations involving the Cauchy stress and the velocity gradient these models are consistent with standard procedures of continuum mechanics. Understanding mathematical properties of governing equations is connected with various types of idealization, some of them lead to studies in unbounded domains. In this paper, we first bring up several characteristic features concerning fluids with pressure dependent viscosity. Then we study three-dimensional flows of a class of fluids with the viscosity depending on the pressure and the shear rate. By means of higher differentiability methods we establish large data existence of a weak solution for the Cauchy problem. This seems to be a first result that analyzes flows of considered fluids in unbounded domains. Even in the context of purely shear rate dependent fluids of a power-law type the result presented here improves some of earlier works.
This paper is devoted to the description of the lack of compactness of $H^1_{rad}(\R^2)$ in the Orlicz space. Our result is expressed in terms of the concentration-type examples derived by P. -L. Lions. The approach that we adopt to establish this characterization is completely different from the methods used in the study of the lack of compactness of Sobolev embedding in Lebesgue spaces and take into account the variational aspect of Orlicz spaces. We also investigate the feature of the solutions of non linear wave equation with exponential growth, where the Orlicz norm plays a decisive role.
We investigate the initial value problem for some energy supercritical semilinear wave equations. We establish local existence in suitable spaces with continuous flow. We also obtain some ill-posedness/weak ill-posedness results. The proof uses the finite speed of propagation and a quantitative study of the associated ODE. It does not require any scaling invariance of the equation.
We investigate the equations of anisotropic incompressible viscous fluids in $\R^3$, rotating around an inhomogeneous vector $B(t, x_1, x_2)$. We prove the global existence of strong solutions in suitable anisotropic Sobolev spaces for small initial data, as well as uniformlocal existence result with respect to the Rossby number in the same functional spaces under the additional assumption that $B=B(t,x_1)$ or $B=B(t,x_2)$. We also obtain the propagation of the isotropic Sobolev regularity using a new refined product law.