arXiv subjects

Mykael Cardoso

Publications and source records attributed to Mykael Cardoso.

16 recordsLinked to original sources

arXiv2026

On the stabilization of $L^2$ and $H^1$ norms for the Zakharov-Kuznetsov equation with damping

In this paper we establish exponential decay results for solutions of the damped $n$-dimensional Zakharov--Kuznetsov equation for $2 \le n \le 3$. More precisely, we prove the exponential decay of the $L^2(\mathbb{R}^n)$ norm when the damping is localized. In addition, when the dissipative mechanism acts on the whole space $\mathbb{R}^n$, we prove the exponential decay of the $H^1(\mathbb{R}^n)$ norm. Our strategy of proof combines a Kato's type smoothing effect, unique continuation and an observability inequality.

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arXiv2026

On the global dynamics and blow-up dichotomy for inhomogeneous coupled nonlinear Schrödinger systems

In this work, we investigate the dynamics of an inhomogeneous coupled nonlinear Schrodinger system with quadratic-type interactions. Such systems arise naturally in nonlinear dynamics and mathematical physics, particularly in nonlinear optics, plasma physics, and wave propagation in inhomogeneous dispersive media. We establish a sharp criterion characterizing the dichotomy between global existence and finite-time blow-up of solutions to the associated initial value problem. This criterion is formulated in terms of conserved quantities, namely mass and energy, measured relative to the ground state solutions of the corresponding elliptic system. The analysis combines variational methods, conservation laws, and sharp Gagliardo-Nirenberg-type inequalities to obtain local and global well-posedness results in both subcritical and intercritical regimes. Our results extend and unify previous studies on single and multi-component nonlinear Schrodinger equations, providing a general analytical framework applicable to a broad class of coupled systems with spatially inhomogeneous nonlinearities and quadratic growth.

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arXiv2024

Minimal mass blow-up solutions for a inhomogeneous NLS equation

We consider the inhomogeneous nonlinear Schrödinger (INLS) equation in $\mathbb{R}^N$ \begin{align}\label{inls} i \partial_t u +Δu +V(x)|u|^{\frac{4-2b}{N}}u = 0, \end{align} where $V(x) = k(x)|x|^{-b}$, with $b>0$. Under suitable assumptions on $k(x)$, we established the threshold for global existence and blow-up and then study the existence and non-existence of minimal mass blow-up solutions.

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arXiv2024

Energy-critical inhomogeneous nonlinear Schrödinger equation with two power-type nonlinearities

We consider the initial value problem for the inhomogeneous nonlinear Schrödinger equation with double nonlinearities (DINLS) \begin{equation*} i \partial_t u + Δu = λ_1 |x|^{-b_1}|u|^{p_1}u + λ_2|x|^{-b_2}|u|^{\frac{4-2b_2}{N-2}}u, \end{equation*} where $λ_1,λ_2\in \mathbb{R}$, $3\leq N<6$ and $0<b_1,b_2<\min\{2,\frac{6-N}{2}\}$. In this paper, we establish global well-posedness results for certain parameter regimes and prove finite-time blow-up phenomena under specific conditions. Our analysis relies on stability theory, energy estimates, and virial identities adapted to the DINLS model.

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arXiv2024

Normalized solutions for INLS equation with critical Hardy-Sobolev type nonlinearities

We are interested in finding prescribed $L^2$-norm solutions to inhomogeneous nonlinear Schrödinger (INLS) equations. For $N\ge 3$ we treat the equation with combined Hardy-Sobolev power-type nonlinearities $$ -Δu+λu=μ|x|^{-b}|u|^{q-2}u+|x|^{-d}|u|^{2^*_{d}-2}u \;\;\mbox{in}\;\; \mathbb{R}^N,\, N\ge 3 $$ where $λ\in\mathbb{R}$, $μ>0$, $0<b,d<2$, $2+(4-2b)/N<q<2+(4-2b)/(N-2)$ and $2^*_{d}= 2(N-d)/(N-2)$ is the Hardy-Sobolev critical exponent, while for $N=2$ we investigate the equation with critical exponential growth \begin{equation}\nonumber \begin{aligned} &-Δu+λu=|x|^{-b}f(u) \;\;\mbox{in}\;\; \mathbb{R}^2 \end{aligned} \end{equation} where the nonlinearity $f(s)$ behaves like $\exp(s^2)$ as $s\to\infty$. We extend the existence results due to Alves-Ji-Miyagaki (Calc. Var. 61, 2022) from $b =d= 0$ to the case $0 < b,d < 2$.

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arXiv2023

On global well-posedness, scattering and other properties for infinity energy solutions to inhomogeneous NLS Equation

In this work, we consider the inhomogeneous nonlinear Schrödinger (INLS) equation in $\mathbb{R}^n$ \begin{align} i\partial_t u + Δu + γ|x|^{-b}|u|^α u = 0, \end{align} where $γ=\pm 1$, and $α$ and $b$ are positive numbers. Our main focus is to estabilish the global well-posedness of the INLS equation in Lorentz spaces for $0<b<2$ and $α<\frac{4-2b}{N-2}$. To achieve this, we use Strichartz estimates in Lorentz spaces $L^{r,q}(\R^n)$ combined with a fixed point argument. Working on Lorentz space setting instead the classical $L^p$ is motivated by the fact that the potential $|x|^{-b}$ does not belong the usual $L^p$-space. As a consequence of the ideas developed here on the global solution study we obtain some other properties for INLS, such as, existence of self-similar solutions, scattering, wave operators and assymptotic stability.

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arXiv2023

Blow-up for the 3D intercritical inhomogeneous NLS with inverse-square potential

In this paper we study the focusing inhomogeneous 3D nonlinear Schrödinger equation with inverse-square potential in the mass-supercritical and energy-subcritical regime. We first establish local well-posedness in $\dot{H}_a^{s_c}\cap \dot{H}_a^1$, with $s_c=3/2-(2-b)/2σ$. Next, we prove the blow-up of the scaling invariant Lebesgue norm for radial solutions and also, with an additional restriction, in the non-radial case.

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arXiv2021

Blow-up of non-radial solutions for the $L^2$ critical inhomogeneous NLS equation

We consider the $L^2$ critical inhomogeneous nonlinear Schrödinger (INLS) equation in $\mathbb{R}^N$ $$ i \partial_t u +Δu +|x|^{-b} |u|^{\frac{4-2b}{N}}u = 0, $$ where $N\geq 1$ and $0<b<2$. We prove that if $u_0\in H^1(\mathbb{R}^N)$ satisfies $E[u_0]<0$, then the corresponding solution blows-up in finite time. This is in sharp contrast to the classical $L^2$ critical NLS equation where this type of result is only known in the radial case for $N\geq 2$.

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arXiv2021

Blow-up solutions of the intercritical inhomogeneous NLS equation: the non-radial case

In this paper we consider the inhomogeneous nonlinear Schrödinger (INLS) equation \begin{align}\label{inls} i \partial_t u +Δu +|x|^{-b} |u|^{2σ}u = 0, \,\,\, x \in \mathbb{R}^N \end{align} with $N\geq 3$. We focus on the intercritical case, where the scaling invariant Sobolev index $s_c=\frac{N}{2}-\frac{2-b}{2σ}$ satisfies $0<s_c<1$. In a previous work, for radial initial data in $\dot H^{s_c}\cap \dot H^1$, we prove the existence of blow-up solutions and also a lower bound for the blow-up rate. Here we extend these results to the non-radial case. We also prove an upper bound for the blow-up rate and a concentration result for general finite time blow-up solutions in $H^1$.

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arXiv2021

Blow-up of radial solutions for the intercritical inhomogeneous NLS equation

We consider the inhomogeneous nonlinear Schrödinger (INLS) equation in $\mathbb{R}^N$ $$i \partial_t u +Δu +|x|^{-b} |u|^{2σ}u = 0,$$ where $N\geq 3$, $0 0$, then $\limsup_{t\rightarrow T^{\ast}}\|u(t)\|_{\dot H^{s_c}}=+\infty$. Moreover, under an additional assumption and recalling that $\dot{H}^{s_c} \subset L^{σ_c}$ with $σ_c=\frac{2Nσ}{2-b}$, we can in fact deduce, for some $γ=γ(N,σ,b)>0$, the following lower bound for the blow-up rate $$c\|u(t)\|_{\dot H^{s_c}}\geq \|u(t)\|_{L^{σ_c}}\geq |\log (T-t)|^γ,\,\,\,\mbox{ as }\,\,\,t\rightarrow T^{\ast}.$$ The proof is based on the ideas introduced for the $L^2$ super critical nonlinear Schrödinger equation in the work of Merle and Raphaël [13] and here we extend their results to the INLS setting.

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arXiv2021

A Virial-Morawetz approach to scattering for the non-radial inhomogeneous NLS

Consider the focusing inhomogeneous nonlinear Schrödinger equation in $H^1(\mathbb{R}^N)$, $$iu_t + Δu + |x|^{-b}|u|^{p-1}u=0,$$ when $b > 0$ and $N \geq 3$ in the intercritical case $0 < s_c <1$. In previous works, the second author, as well as Farah, Guzmán and Murphy, applied the concentration-compactness approach to prove scattering below the mass-energy threshold for radial and non-radial data. Recently, the first author adapted the Dodson-Murphy approach for radial data, followed by Murphy, who proved scattering for non-radial solutions in the 3d cubic case, for $b<1/2$. This work generalizes the recent result of Murphy, allowing a broader range of values for the parameters $p$ and $b$, as well as allowing any dimension $N \geq 3$. It also gives a simpler proof for scattering nonradial, avoiding the Kenig-Merle road map. We exploit the decay of the nonlinearity, which, together with Virial-Morawetz-type estimates, allows us to drop the radial assumption.

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arXiv2020

Global well-posedness and critical norm concentration for inhomogeneous biharmonic NLS

We consider the inhomogeneous biharmonic nonlinear Schrödinger (IBNLS) equation in $\mathbb{R}^N$, $$i \partial_t u +Δ^2 u -|x|^{-b} |u|^{2σ}u = 0,$$ where $σ>0$ and $b>0$. We first study the local well-posedness in $\dot H^{s_c}\cap \dot H^2 $, for $N\geq 5$ and $0<s_c<2$, where $s_c=\frac{N}{2}-\frac{4-b}{2σ}$. Next, we established a Gagliardo-Nirenberg type inequality in order to obtain sufficient conditions for global existence of solutions in $\dot H^{s_c}\cap \dot H^2$ with $0\leq s_c<2$. Finally, we study the phenomenon of $L^{σ_c}$-norm concentration for finite time blow up solutions with bounded $\dot H^{s_c}$-norm, where $σ_c=\frac{2Nσ}{4-b}$. Our main tool is the compact embedding of $\dot L^p\cap \dot H^2$ into a weighted $L^{2σ+2}$ space, which may be seen of independent interest.

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arXiv2020

Scattering below the ground state for the intercritical non-radial inhomogeneous NLS

We consider the focusing inhomogeneous nonlinear Schrödinger equation \[ i\partial_t u + Δu + |x|^{-b}|u|^αu = 0\quad\text{on}\quad\mathbb{R}\times\mathbb{R}^N, \] with $N\geq 2$, $0<b<\min\{\tfrac{N}{2},2\}$, and $\tfrac{4-2b}{N}<α<\tfrac{4-2b}{N-2}$. These constraints make the equation mass-supercritical and energy-subcritical. We extend the results of Farah-Guzmán and Miao-Murphy-Zheng and prove scattering below the ground state with general initial data.

math.AP↗

arXiv2020

On well-posedness and concentration of blow-up solutions for the intercritical inhomogeneous NLS equation

We consider the focusing inhomogeneous nonlinear Schrödinger (INLS) equation in $\mathbb{R}^N$ $$i \partial_t u +Δu + |x|^{-b} |u|^{2σ}u = 0,$$ where $N\geq 2$ and $σ$, $b>0$. We first obtain a small data global result in $H^1$, which, in the two spatial dimensional case, improves the third author result in [22] on the range of $b$. For $N\geq 3$ and $\frac{2-b}{N}<σ<\frac{2-b}{N-2}$, we also study the local well posedness in $\dot H^{s_c}\cap \dot H^1 $, where $s_c=\frac{N}{2}-\frac{2-b}{2σ}$. Sufficient conditions for global existence of solutions in $\dot H^{s_c}\cap \dot H^1$ are also established, using a Gagliardo-Nirenberg type estimate. Finally, we study the $L^{σ_c}-$norm concentration phenomenon, where $σ_c=\frac{2Nσ}{2-b}$, for finite time blow-up solutions in $\dot H^{s_c}\cap \dot H^1$ with bounded $\dot H^{s_c}-$norm. Our approach is based on the compact embedding of $\dot H^{s_c}\cap \dot H^1$ into a weighted $L^{2σ+2}$ space.

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arXiv2020

Blow up and scattering criteria above the threshold for the focusing inhomogeneous nonlinear Schrödinger equation

We consider the inhomogeneous nonlinear Schrödinger equation (INLS) in $\mathbb{R}^N$, $N \geq 1$, $$i \partial_t u + Δu + |x|^{-b} |u|^{p-1}u = 0,$$ with finite-variance initial data $u_0 \in H^1(\mathbb{R}^N)$. We extend the dichotomy between scattering and blow-up for solutions above the mass-energy threshold (and with arbitrarily large energy). We also show other two blow-up criteria, wich are valid in any mass-supercritical setting, given there is local well-posedness.

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arXiv2018

On the critical norm concentration for the inhomogeneous nonlinear Schrödinger equation

We consider the inhomogeneous nonlinear Schrödiger equation (INLS) in $\mathbb{R}^N$ $$i \partial u_t + Δu + |x|^{-b} |u|^{2σ}u = 0,$$ and show the $L^2$-norm concentration for the finite time blow-up solutions in the $L^2$-critical case, $σ=\frac{2-b}{N}$. Moreover, we provide an alternative for the classification of minimal mass blow-up solutions first proved by Genoud and Combet [4]. For the case $\frac{2-b}{N} < σ< \frac{2-b}{N-2}$, we show results regarding the $L^p$-critical norm concentration, generalizing the argument of Holmer and Roudenko [16] to the INLS setting.

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