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We consider the two-dimensional nonlinear Schr\"odinger equation with point interaction and we establish a local well-posedness theory, including blow-up alternative and continuous dependence on the initial data in the energy space. We provide a proof by employing a Kato's method along with Hardy inequalities with logarithmic correction. Moreover, we establish finite time blow-up for solutions with positive energy and infinite variance., Comment: 22 pages, submitted

In this paper, we consider solutions to the following fourth order anisotropic nonlinear Schr\"odinger equation in $\R \times \R^2$, $$ \left\{ \begin{aligned} &\textnormal{i}\partial_t\psi+\partial_{xx} \psi-\partial_{yyyy} \psi +|\psi|^{p-2} \psi=0, \\ &\psi(0)=\psi_0 \in H^{1,2}(\R^2), \end{aligned} \right. $$ where $p>2$. First we prove the local/global well-posedness and blowup of solutions to the Cauchy problem for the anisotropic nonlinear Schr\"odinger equation. Then we establish the existence, axial symmetry, exponential decay and orbital stability/instability of standing waves to the anisotropic nonlinear Schr\"odinger equation. The pictures are considerably different from the ones for the isotropic nonlinear Schr\"odinger equations. The results are easily extendable to the higher dimensional case., Comment: 22 pages

We study the existence of positive solutions of a particular elliptic system in $\mathbb{R}^3$ composed of two coupled non linear stationary Schr\"odinger equations (NLSEs), that is $-\epsilon^2 \Delta u + V(x) u= h_v(u,v), - \epsilon^2 \Delta v + V(x) v=h_u (u,v)$. Under certain hypotheses on the potential $V$ and the non linearity $h$, we manage to prove that there exists a solution $(u_\epsilon,v_\epsilon)$ that decays exponentially with respect to local minima points of the potential and whose energy tends to concentrate around these points, as $\epsilon \to 0$. We also estimate this energy in terms of particular ground state energies. This work follows closely what is done in https://doi.org/10.1007/s00526-007-0103-z , although here we consider a more general non linearity and we restrict ourselves to the case where the domain is $\mathbb{R}^3$., Comment: This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in AIP Conf.Proc. 5 April 2022; 2459 (1):030003 and may be found at https://doi.org/10.1063/5.0084041

The work deals with the Ericksen-Leslie model for nematic liquid crystals on the whole space, the half-space and on exterior domains with smooth boundary. The crystal orientation is described by a unit vector that is a small perturbation of a fixed constant vector. We prove, through a combination of energy method with dispersive a priori estimates, a local existence and a global existence for small initial data by a contraction argument asking low regularity assumptions on the initial condition., Comment: 38 pages

We study the perturbed Sobolev space $H^{1,r}_\alpha$, $r \in (1,\infty),$ associated with singular perturbation $\Delta_\alpha$ of Laplace operator in Euclidean space of dimension $2.$ The main results give the possibility to extend the $L^2$ theory of perturbed Sobolev space to the $L^r$ case. When $r \in (2,\infty)$ we have appropriate representation of the functions in $H^{1,r}_\alpha$ in regular and singular part. An application to local well - posedness of the NLS associated with this singular perturbation in the mass critical and mass supercritical cases is established too., Comment: 20 pages

In this manuscript, a sharp lifespan estimate of solutions to semilinear classical damped wave equation is investigated in one dimensional case when the Fourier 0th moment of sum of initial position and speed is $0$. Especially, it is shown that the behavior of lifespan changes with $p=3/2$ with respect to the size of the initial data.

We study the existence, the stability and the non-degeneracy of normalized standing-waves solutions to a one dimensional non-linear Schr\"odinger equation. The non-linearity belongs to a class of algebraic functions appropriately defined. We can show that for some of these non-linearities one can observe the existence of degenerate minima, and the multiplicity of positive, radially symmetric minima having the same mass and the same energy. We also prove the stability of the ground-state and the stability of normalized standing-waves whose profile is a minimum of the energy constrained to the mass., Comment: 19 pages, 5 figures

The work deals with the Ericksen-Leslie System for nematic liquid crystals on the whole space. In our work we suppose the initial condition of the orientation field stays on an arc connecting two fixed orthogonal vectors on the unit sphere. Thanks to this geometric assumption, we prove through energy a priori estimates the local existence and the global existence for small initial data of a solution in low regularity Sobolev spaces., Comment: 40 p

In this manuscript, a sharp lifespan estimate of solutions to semilinear classical damped wave equation is investigated in one dimensional case, when the sum of initial position and speed is $0$ pointwisely. Especially, an extension of lifespan is shown in this case. Moreover, existence of some global solutions are obtained by a direct computation., Comment: 11 pages

We consider the half-wave equation $iu_t=Du-|u|u$ in two dimensions. For the initial data $u_0(x)\in H^{s}(\mathbb{R}^2)$, $s\in\left(\frac{3}{4},1\right)$, we obtain the non-radial ground state mass blow-up solutions with the blow-up speed $\|D^{\frac{1}{2}}u(t)\|_{L^2}\sim\frac{1}{|t|}$ as $t\to 0^-$., Comment: 28 pages. arXiv admin note: substantial text overlap with arXiv:2007.15370

We study a class of two-dimensional non-linear Schr\"odinger equations with point-like singular perturbation and Hartree non-linearity. The point-like singular perturbation of the free Laplacian induces appropriate perturbed Sobolev spaces that are necessary for the study of ground states and evolution flow. We include in our treatment both mass sub-critical and mass critical Hartree non-linearities. Our analysis is two-fold: we establish existence, symmetry, and regularity of ground states, and we demonstrate the well-posedness of the associated Cauchy problem in the singular perturbed energy space. The first goal, unlike other treatments emerging in parallel with the present work, is achieved by a non-trivial adaptation of the standard properties of Schwartz symmetrisation for the modified Weinstein functional. This produces, among others, modified Gagliardo-Nirenberg type inequalities that allow to efficiently control the non-linearity and obtain well-posedness by energy methods. The evolution flow is proved to be global in time in the defocusing case, and in the focusing and mass sub-critical case. It is also global in the focusing and mass critical case, for initial data that are suitably small in terms of the best Gagliardo-Nirenberg constant.

We prove compactness of the embeddings in Sobolev spaces for fractional super and sub harmonic functions with radial symmetry. The main tool is a pointwise decay for radially symmetric functions belonging to a function space defined by finite homogeneous Sobolev norm together with finite $L^2$ norm of the Riesz potentials. As a byproduct we prove also existence of maximizers for the interpolation inequalities in Sobolev spaces for radially symmetric fractional super and sub harmonic functions., Comment: 26 pages, 1 figure

In this paper, we are concerned with solutions to the Cauchy problem for Chern-Simons-Schr\"odinger equations in the mass supercritical case. First we establish the local well-posedness of solutions in the radial space. Then we consider scattering versus blow-up dichotomy for radial data below the ground state threshold., Comment: 33 pages, no figure

In this work we study the global existence for 3d semilinear wave equation with non-negative potential satisfying generic decay assumptions. In the supercritical case we establish the small data global existence result. The approach is based on appropriate conformal energy estimate in combination with Hardy inequality for conformal energy on space - like surfaces., Comment: 22 pages, 2 figures

We consider the following nonlinear Schr\"{o}dinger equation with the double $L^2$-critical nonlinearities \begin{align*} iu_t+\Delta u+|u|^\frac{4}{3}u+\mu\left(|x|^{-2}*|u|^2\right)u=0\ \ \ \text{in $\mathbb{R}^3$,} \end{align*} where $\mu>0$ is small enough. Our first goal is to prove the existence and the non-degeneracy of the ground state $Q_{\mu}$. In particular, we develop an appropriate perturbation approach to prove the radial non-degeneracy property and then obtain the general non-degeneracy of the ground state $Q_{\mu}$. We then show the existence of finite time blowup solution with minimal mass $\|u_0\|_{L^2}=\|Q_{\mu}\|_{L^2}$. More precisely, we construct the minimal mass blowup solutions that are parametrized by the energy $E_{\mu}(u_0)>0$ and the momentum $P_{\mu}(u_0)$. In addition, the non-degeneracy property plays crucial role in this construction., Comment: 39pages. arXiv admin note: text overlap with arXiv:1001.1627, arXiv:1203.2476 by other authors

We study existence and stability properties of ground-state standing waves for two-dimensional nonlinear Schr\"odinger equation with a point interaction and a focusing power nonlinearity. The Schr\"odinger operator with a point interaction describes a one-parameter family of self-adjoint realizations of the Laplacian with delta-like perturbation. The perturbed Laplace operator always has a unique simple negative eigenvalue. We prove that if the frequency of the standing wave is close to the negative eigenvalue, it is stable. Moreover, if the frequency is sufficiently large, we have the stability in the $L^2$-subcritical or critical case, while the instability in the $L^2$-supercritical case., Comment: 29pages

We consider the nonlinear Schr\"odinger equation with combined nonlinearities, where the leading term is an intracritical focusing power-type nonlinearity, and the perturbation is given by a power-type defocusing one. We completely answer the question wether the ground state energy, which is a threshold between global existence and formation of singularities, is achieved. For any prescribed mass, for mass-supercritical or mass-critical defocusing perturbations, the ground state energy is achieved by a radially symmetric and decreasing solution to the associated stationary equation. For mass-subcritical perturbations, we show the existence of a critical prescribed mass, precisely the mass of the unique, static, positive solution to the associated elliptic equation, such that the ground state energy is achieved for any mass equal or smaller than the critical one. Moreover, the ground state energy is not achieved for mass larger than the critical one. As a byproduct of the variational characterization of the ground state energy, we prove the existence of blowing-up solutions in finite time, for any energy below the ground state energy threshold., Comment: 30 pages, 2 figures. Submitted paper

We consider the half-wave equation $i u_t=Du-|u|^{\frac{2}{3}}u$ in three dimension and in the mass critical. For initial data $u(t_0,x)=u_0(x)\in H^{1/2}_{rad}(\mathbb{R}^3)$ with radial symmetry, we construct a new class of minimal mass blowup solutions with the blow up rate $\|D^{\frac{1}{2}}u(t)\|_2\sim\frac{C(u_0)}{|t|^{\frac{1}{4}}}$ as $t\rightarrow0^-$., Comment: arXiv admin note: substantial text overlap with arXiv:2007.15370

We study 1D NLS with non-gauge invariant quadratic nonlinearity on the torus. The Cauchy problem admits trivial global solutions which are constant with respect to space. The non-existence of global solutions also has been studied only by focusing on the behavior of the Fourier $0$ mode of solutions. However, the earlier works are not sufficient to obtain the precise criteria for the global existence for the Cauchy problem. In this paper, the exact criteria for the global existence of $L^2$ solutions is shown by studying the interaction between the Fourier $0$ mode and oscillation of solutions. Namely, $L^2$ solutions are shown a priori not to exist globally if they are different from the trivial ones., Comment: 11 pages, no figure

We consider the half-wave equation with mass critical in two dimension \begin{eqnarray*} \begin{cases} iu_t=Du-|u|u,\,\,\, \\ u(0,x)=u_0(x), \end{cases} \end{eqnarray*} First, we prove the existence of a family of traveling solitary waves. We then show the existence of finite-time blowup solutions with minimal mass $\|u_0\|_2=\|Q\|_2$, where $Q$ is the ground state solution of equation $DQ+Q=Q^2$.

Local and global well - posedness of the solution to the two space dimensional Dirac equation with Hartree type nonlinearity is established with the initial datum in the space $H^s(\mathbb{R}^2, \mathbb{C}^2)$ with $s >0.$.

We consider the generalized Choquard equation describing trapped electron gas in 3 dimensional case. The study of orbital stability of the energy minimizers (known as ground states) depends essentially in the local uniqueness of these minimizers. In equivalent way one can optimize the Gagliardo--Nirenberg inequality subject to the constraint fixing the $L^2$ norm. The uniqueness of the minimizers for the case $p=2$, i.e. for the case of Hartree--Choquard is well known. The main difficulty for the case $p > 2$ is connected with possible lack of control on the $L^p$ norm of the minimizers., Comment: Some misprints have been corrected

In this paper, we consider the Cauchy problem for the semilinear damped wave equation on the Heisenberg group with power nonlinearity. We prove that the critical exponent is the Fujita exponent $p_{\mathrm{Fuj}}(\mathscr{Q}) = 1+2 / \mathscr{Q}$, where $\mathscr{Q}$ is the homogeneous dimension of the Heisenberg group. On the one hand, we will prove the global existence of small data solutions for $p >p_{\mathrm{Fuj}}(\mathscr{Q})$ in an exponential weighted energy space. On the other hand, a blow-up result for $1 < p \leq p_{\mathrm{Fuj}}(\mathscr{Q})$ under certain integral sign assumptions for the Cauchy data by using the test function method.

In this paper we consider the semilinear Cauchy problem for the heat equation with power nonlinearity in the Heisenberg group $\mathbf{H}_n$. The heat operator is given in this case by $\partial_t-\Delta_H$, where $\Delta_H$ is the so-called sub-Laplacian on $\mathbf{H}_n$. We prove that the Fujita exponent $1 + 2/Q$ is critical, where $Q = 2n + 2$ is the homogeneous dimension of $\mathbf{H}_n$. Furthermore, we prove sharp lifespan estimates for local in time solutions in the subcritical case and in the critical case. In order to get the upper bound estimate for the lifespan (especially, in the critical case) we employ a revisited test function method developed recently by Ikeda-Sobajima. On the other hand, to find the lower bound estimate for the lifespan we prove a local in time result in weighted $L^\infty$ space.

The purpose in this paper is to prove end point Strichartz estimates for the Schr\"odinger equation in the exterior domain of a generic non-trapping obstacle in the case $n \geq 3.$ In the case $n=2$ we have the same range of Strichartz estimates as in the free case. In this version we corrected some misprints from the previous one., Comment: We found an essential gap in the proof of the main result

We consider the scattering problem for the nonlinear Schr\"{o}dinger equation with a potential in two space dimensions. Appropriate resolvent estimates are proved and applied to estimate the operator $A(s)$ appearing in commutator relations. The equivalence between the operators $\left(-\Delta_{V}\right)^{\frac{s}{2}}$ and $\left(-\Delta \right)^{\frac{s}{2}}$ in $L^{2}$ norm sense for $0\leq s <1$ is investigated by using free resolvent estimates and Gaussian estimates for the heat kernel of the Schr\"{o}dinger operator $-\Delta_{V}$. Our main result guarantees the global existence of solutions and time decay of the solutions assuming initial data have small weighted Sobolev norms. Moreover, the global solutions obtained in the main result scatter.

A class of self-similar solutions to the derivative nonlinear Schr\"odinger equations is studied. Especially, the asymptotics of profile functions are shown to posses a logarithmic phase correction. This logarithmic phase correction is obtained from the nonlinear interaction of profile functions. This is a remarkable difference from the pseudo-conformally invariant case, where the logarithmic correction comes from the linear part of the equations of the profile functions., Comment: 16 pages, no figures

We are studying possible interaction of damping coefficients in the subprincipal part of the linear 3D wave equation and their impact on the critical exponent of the corresponding nonlinear Cauchy problem with small initial data. The main new phenomena is that certain relation between these coefficients may cause very strong jump of the critical Strauss exponent in 3D to the critical 5D Strauss exponent for the wave equation without damping coefficients., Comment: 16 pages

We consider nonlinear half-wave equations with focusing power-type nonlinearity $$ i \pt_t u = \sqrt{-\Delta} \, u - |u|^{p-1} u, \quad \mbox{with $(t,x) \in \R \times \R^d$} $$ with exponents $1 < p < \infty$ for $d=1$ and $1 < p < (d+1)/(d-1)$ for $d \geq 2$. We study traveling solitary waves of the form $$ u(t,x) = e^{i\omega t} Q_v(x-vt) $$ with frequency $\omega \in \R$, velocity $v \in \R^d$, and some finite-energy profile $Q_v \in H^{1/2}(\R^d)$, $Q_v \not \equiv 0$. We prove that traveling solitary waves for speeds $|v| \geq 1$ do not exist. Furthermore, we generalize the non-existence result to the square root Klein--Gordon operator $\sqrt{-\DD+m^2}$ and other nonlinearities. As a second main result, we show that small data scattering fails to hold for the focusing half-wave equation in any space dimension. The proof is based on the existence and properties of traveling solitary waves for speeds $|v| < 1$. Finally, we discuss the energy-critical case when $p=(d+1)/(d-1)$ in dimensions $d \geq 2$., Comment: 17 pages

We consider the quartic focusing Half Wave equation (HW) in one space dimension. We show first that that there exist traveling wave solutions with arbitrary small $H^{\frac 12}(\R)$ norm. This fact shows that small data scattering is not possible for (HW) equation and that below the ground state energy there are solutions whose energy travels as a localised packet and which preserve this localisation in time. This behaviour for (HW) is in sharp contrast with classical NLS in any dimension and with fractional NLS with radial data. The second result addressed is the non existence of traveling waves moving at the speed of light. The main ingredients of the proof are commutator estimates and a careful study of spatial decay of traveling waves profile using the harmonic extension to the upper half space., Comment: 20 pages

We study the Cauchy problem for the half Ginzburg-Landau-Kuramoto (hGLK) equation with the second order elliptic operator having rough coefficients and potential type perturbation. The blow-up of solutions for hGLK equation with non-positive nonlinearity is shown by an ODE argument. The key tools in the proof are appropriate commutator estimates and the essential self-adjointness of the symmetric uniformly elliptic operator with rough metric and potential type perturbation., Comment: 22 pages, no figures

The local and global existence of the Cauchy problem for semilinear heat equations with small data is studied in the weighted $L^\infty (\mathbb R^n)$ framework by a simple contraction argument. The contraction argument is based on a weighted uniform control of solutions related with the free solutions and the first iterations for the initial data of negative power., Comment: We withdraw this manuscript because a similar argument has already been discussed by T. Cazenave, F. Dickstein, and F.B. Weissler, Universal solutions of a nonlinear heat equation on $\mathbb R^N$, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2 (2003), no. 1, 77 -- 117

We study the $p$-Choquard equation in 3-dimensional case and establish existence and uniqueness of ground states for the corresponding Weinstein functional. For proving the uniqueness of ground states, we use the radial symmetry to transform the equation into an ordinary differential system, and applying the Pohozaev identities and Gronwall lemma we show that any two Weinstein minimizers coincide., Comment: 13

This paper considers Hamiltonians with localised potentials and gives a variational characterisation of resonant coupling parameters, which allow to provide estimates for the first resonant parameter and in turn also to provide bounds for resonant free regions. As application we provide a constructive approach to calculate the first resonant parameter for Yukawa type potentials in $\mathbb R^3$., Comment: 16 pages

We qualify a relevant range of fractional powers of the so-called Hamiltonian of point interaction in three dimensions, namely the singular perturbation of the negative Laplacian with a contact interaction supported at the origin. In particular we provide an explicit control of the domain of such a fractional operator and of its decomposition into regular and singular parts. We also qualify the norms of the resulting singular fractional Sobolev spaces and their mutual control with the corresponding classical Sobolev norms.