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Luccas Campos

Publications and source records attributed to Luccas Campos.

At least 19 recordsLinked to original sources
arXiv2026

Existence and equicontinuity of solutions to the Kirchhoff--Pohozaev wave equation

It was recently discovered by Boiti--Manfrin that a variant of the Kirchhoff wave equation introduced by Pohozaev admits infinitely many conserved quantities. In this paper, we obtain explicit formulae for such conserved quantities, as well as introduce a generating function for them that is coercive. These tools are then employed to establish a priori bounds, the propagation of equicontinuity, global well-posedness in $\mathcal H^s$ for $s\geq \frac32$, and the existence of global $C_t \mathcal H^1$ solutions.

math.AP↗
arXiv2026

Large data scattering for the defocusing $k$-dispersion generalized Benjamin-Ono equation in the energy space

We study the defocusing $k$-dispersion generalized Benjamin-Ono equation. For every even integer $k\geq 4$, we prove that solutions with initial data in the energy space $H^{\fracα{2}}$ are global in time and scatter. The proof combines the concentration-compactness-rigidity method of Kenig and Merle with techniques based on the Caffarelli-Silvestre extension and Tao's monotonicity formula adapted to the fractional dispersion setting.

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arXiv2026

Determination of radial nonlocal nonlinearities from the scattering map

We show that the small-data scattering map uniquely determines the nonlinearity for a class of nonlinear Schrödinger equations with radial, Hartree-type nonlinearities. Our assumptions on the convolution kernel require only a mild decay condition at infinity and permit a locally integrable singularity at the origin.

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arXiv2026

Threshold solutions for the $3d$ cubic INLS: the energy-subcritical case

We revisit the work [L. Campos and J. Murphy, SIAM J. Math. Anal., 55 (2023), pp. 3807--3843], which classified the dynamics of $H^1$ solutions at the ground state threshold for cubic inhomogeneous nonlinear Schrödinger equations of the form $i\partial_t u + Δu + |x|^{-b}|u|^2 u = 0$ in the range $b\in(0,\tfrac12)$. By modifying the modulation analysis and using Strichartz estimates in place of pointwise bounds, we extend the result to the full energy-subcritical range $b\in(0,1)$. This strategy is expected to carry over to other dispersive equations with singular potentials.

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arXiv2026

Threshold solutions for the $3d$ cubic INLS: the energy-critical case

We study the energy-critical $3d$ cubic inhomogeneous NLS equation $i\partial_t u + Δu + |x|^{-1}|u|^2 u=0$. In this work, we prove the existence of special solutions $W^\pm$ with energy equal to that of the ground state $W$ and use these solutions to characterize the behavior of solutions at the ground state energy. The singular factor $|x|^{-1}$ in the nonlinearity significantly limits the smoothness of the ground state and prompts a novel approach to the modulation analysis.

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arXiv2024

Sharp well-posedness for the $k$-dispersion generalized Benjamin-Ono equations: Short and long time results

We consider the $k$-dispersion generalized Benjamin-Ono ($k$-DGBO) equations. For nonlinearities with power $k \geq 4$, we establish local and global well-posedness results for the associated initial value problem (IVP) in both the critical and subcritical regimes, addressing sharp regularity in homogeneous and inhomogeneous Sobolev spaces. Additionally, our method enables the formulation of a scattering criterion and a scattering theory for small data. We also investigate the case $k = 3$ via frequency-restricted estimates, obtaining local well-posedness results for the IVP associated with the $3$-DGBO equation and generalizing the existing results in the literature for the whole subcritical range. For higher dispersion, these local results can be extended globally even for rough data, particularly for initial data in Sobolev spaces with negative indices. As a byproduct, we derive new nonlinear smoothing estimates.

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arXiv2024

Sharp well-posedness and ill-posedness results for the inhomogeneous NLS equation

We consider the initial value problem associated to the inhomogeneous nonlinear Schrö\-din\-ger equation, \begin{equation} iu_t + Δu +μ|x|^{-b}|u|^αu=0, \quad u_0\in H^s(\mathbb R^N) \text{ or } u_0 \in\dot H ^s(\mathbb R^N), \end{equation} with $μ=\pm 1$, $b > 0$, $s\geq 0$ and $0 < α\leq \frac{4-2b}{N-2s}$. By means of an adapted version of the fractional Leibniz rule, we prove new local well-posedness results in Sobolev spaces for a large range of parameters. We also prove an ill-posedness result for this equation, through a delicate analysis of the associated Duhamel operator.

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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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arXiv2022

Threshold solutions for cubic Schrödinger systems

We consider the following Scrödinger system $$\begin{cases}\displaystyle i\partial_t u + Δu +(|u|^2+β|v|^2) u= 0, \\ \displaystyle i\partial_t v + Δv +(|v|^2+β|u|^2) v = 0,\end{cases}$$ with initial data $(u_0,v_0) \in H^1(\mathbb{R} ^3)\times H^1(\mathbb{R}^3)$ at the so-called \textit{mass-energy threshold}, i.e., such that %$\mathcal{ME}(u_0,v_0) = 1$. $M(u_0,v_0)E(u_0,v_0) = M(ϕ,ψ)E(ϕ,ψ)$, where $(ϕ,ψ)$ is a ground state. For a suitable range of values of $β>0$, we show the existence of special solutions to this system, which converge to a standing wave solution in one time direction, and either blows up or scatters in the opposite direction. Moreover, we classify general solutions at the ground state, showing a rigidity result regarding the possible long-time behaviors that might occur. Our results do not rely on the uniqueness of the corresponding ground state: indeed, the main results hold even in the case where the Weinstein functional is known to have more than one optimizer.

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arXiv2022

Averaging for the dispersion-managed NLS

We establish global-in-time averaging for the $L^2$-critical dispersion-managed nonlinear Schrödinger equation in the fast dispersion management regime. In particular, in the case of nonzero average dispersion, we establish averaging with any subcritical data, while in the case of a strictly positive dispersion map, we obtain averaging for data in $L^2$.

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arXiv2022

Threshold solutions for the intercritical inhomogeneous NLS

We consider the focusing inhomogeneous nonlinear Schrödinger equation in $H^1(\mathbb{R}^3)$, \begin{equation} i\partial_t u + Δu + |x|^{-b}|u|^{2}u=0,{equation} where $0 < b <\tfrac{1}{2}$. Previous works have established a blowup/scattering dichotomy below a mass-energy threshold determined by the ground state solution $Q$. In this work, we study solutions exactly at this mass-energy threshold. In addition to the ground state solution, we prove the existence of solutions $Q^\pm$, which approach the standing wave in the positive time direction, but either blow up or scatter in the negative time direction. Using these particular solutions, we classify all possible behaviors for threshold solutions. In particular, the solution either behaves as in the sub-threshold case, or it agrees with $e^{it}Q$, $Q^+$, or $Q^-$ up to the symmetries of the equation.

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arXiv2021

Scattering for the non-radial inhomogenous biharmonic NLS equation

We consider the focusing inhomogeneous biharmonic nonlinear Schrödinger equation in $H^2(\mathbb{R}^N)$, \begin{equation} iu_t + Δ^2 u - |x|^{-b}|u|^αu=0 \end{equation} when $b > 0$ and $N \geq 5$. We first obtain a small data global result in $H^2$, which, in the five-dimensional case, improves a previous result from Pastor and the second author. In the sequel, we show the main result, scattering below the mass-energy threshold in the intercritical case, that is, $\frac{8-2b}{N} < α<\frac{8-2b}{N-4}$, without assuming radiality of the initial data. The proof combines the decay of the nonlinearity with Virial-Morawetz-type estimates to avoid the radial assumption, allowing for a much simpler proof than the Kenig-Merle roadmap.

math.AP↗
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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arXiv2021

On the inhomogeneous NLS with inverse-square potential

We consider the inhomogeneous nonlinear Schrödinger equation with inverse-square potential in $\mathbb{R}^N$ $$ i u_t + \mathcal{L}_a u+λ|x|^{-b}|u|^αu = 0,\;\;\mathcal{L}_a=Δ-\frac{a}{|x|^2}, $$ where $λ=\pm1$, $α,b>0$ and $a>-\frac{(N-2)^2}{4}$. We first establish sufficient conditions for global existence and blow-up in $H^1_a(\mathbb{R}^N)$ for $λ=1$, using a Gagliardo-Nirenberg-type estimate. In the sequel, we study local and global well-posedness in $H^1_a(\mathbb{R}^N)$ in the $H^1$-subcritical case, applying the standard Strichartz estimates combined with the fixed point argument. The key to do that is to establish good estimates on the nonlinearity. Making use of these estimates, we also show a scattering criterion and construct a wave operator in $H^1_a(\mathbb{R}^N)$, for the mass-supercritical and energy-subcritical case.

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arXiv2020

Threshold solutions for the nonlinear Schrödinger equation

We study the focusing NLS equation in $\mathbb{R}^N$ in the mass-supercritical and energy-subcritical (or intercritical) regime, with $H^1$ data at the mass-energy threshold $ \mathcal{ME}(u_0)=\mathcal{ME}(Q)$, where $Q$ is the ground state. Previously, Duyckaerts-Merle studied the behavior of threshold solutions in the $H^1$-critical case, in dimensions $N = 3, 4, 5$, later generalized by Li-Zhang for higher dimensions. In the intercritical case, Duyckaerts-Roudenko studied the threshold problem for the 3d cubic NLS equation. In this paper, we generalize the results of Duyckaerts-Roudenko for any dimension and any power of the nonlinearity for the entire intecritical range. We show the existence of special solutions, $Q^\pm$, besides the standing wave $e^{it}Q$, which exponentially approach the standing wave in the positive time direction, but differ in its behavior for negative time. We classify all solutions at the threshold level, showing either blow-up occurs in finite (positive and negative) time, or scattering in both time directions, or the solution is equal to one of the three special solutions above, up to symmetries. Our proof extends to the $H^1$-critical case, thus, giving a different and more unified approach than the Li-Zhang result. These results are obtained by studying the linearized equation around the standing wave and some tailored approximate solutions to the NLS equation. We establish important decay properties of functions associated to the spectrum of the linearized Schrödinger operator, which, in combination with modulational stability and coercivity for the linearized operator on special subspaces, allows us to use a fixed-point argument to show the existence of special solutions. Finally, we prove the uniqueness by studying exponentially decaying solutions to a sequence of linearized equations.

math.AP↗
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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