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Lassaad Aloui

Publications and source records attributed to Lassaad Aloui.

6 recordsLinked to original sources
arXiv2023

Asymptotic behavior and life-span estimates for the damped inhomogeneous nonlinear Schrödinger equation

We are interested in the behavior of solutions to the damped inhomogeneous nonlinear Schrödinger equation $ i\partial_tu+Δu+μ|x|^{-b}|u|^αu+iau=0$, $μ\in\mathbb{C} $, $b>0$, $a \in \mathbb{C}$ such that $\Re \textit{e}(a) \geq 0$, $α>0$. We establish lower and upper bound estimates of the life-span. In particular for $a\geq 0$, we obtain explicit values $a_*,\; a^*$ such that if $a a^*,$ global existence holds. Also, we prove scattering results with precise decay rates for large damping. Some of the results are new even for $b=0.$

math.AP↗
arXiv2021

Global existence and scattering for the inhomogeneous nonlinear Schrödinger equation

In this paper we consider the inhomogeneous nonlinear Schrödinger equation $i\partial_t u +Δu=K(x)|u|^αu,\, u(0)=u_0\in H^s({\mathbb R}^N),\, s=0,\,1,$ $N\geq 1,$ $|K(x)|+|x|^s|\nabla^sK(x)|\lesssim |x|^{-b},$ $0<b<\min(2,N-2s),$ $0<α<{(4-2b)/(N-2s)}$. We obtain novel results of global existence for oscillating initial data and scattering theory in a weighted $L^2$-space for a new range $α_0(b)<α<(4-2b)/N$. The value $α_0(b)$ is the positive root of $Nα^2+(N-2+2b)α-4+2b=0,$ which extends the Strauss exponent known for $b=0$. Our results improve the known ones for $K(x)=μ|x|^{-b}$, $μ\in \mathbb{C}$ and apply for more general potentials. In particular, we show the impact of the behavior of the potential at the origin and infinity on the allowed range of $α$. Some decay estimates are also established for the defocusing case. To prove the scattering results, we give a new criterion taking into account the potential $K$.

math.AP↗
arXiv2015

Energy decay for linear dissipative wave equations in exterior domains

In earlier works, we have shown the uniform decay of the local energy of the damped wave equation in exterior domain when the damper is spatially localized near captive rays. In order to have uniform decay of the total energy, the damper has also to act at space infinity. In this work, we establish uniform decay of both the local and global energies. The rates of decay turns out to be the same as those for the heat equation, which shows that an effective damper at space infinity strengthens the parabolic structure in the equation.

math.AP↗