arXiv subjects
Publications and source records attributed to Vladimir Georgiev.
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.
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 $t\to 0^-$.
We study a class of two-dimensional non-linear Schrödinger 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.
In this paper, we are concerned with solutions to the Cauchy problem for Chern-Simons-Schrödinger 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.
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.
We consider the following nonlinear Schrödinger equation with the double $L^2$-critical nonlinearities \begin{align*} iu_t+Δu+|u|^\frac{4}{3}u+μ\left(|x|^{-2}*|u|^2\right)u=0\ \ \ \text{in $\mathbb{R}^3$,} \end{align*} where $μ>0$ is small enough. Our first goal is to prove the existence and the non-degeneracy of the ground state $Q_μ$. 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 $Q_μ$. We then show the existence of finite time blowup solution with minimal mass $\|u_0\|_{L^2}=\|Q_μ\|_{L^2}$. More precisely, we construct the minimal mass blowup solutions that are parametrized by the energy $E_μ(u_0)>0$ and the momentum $P_μ(u_0)$. In addition, the non-degeneracy property plays crucial role in this construction.
We study existence and stability properties of ground-state standing waves for two-dimensional nonlinear Schrödinger equation with a point interaction and a focusing power nonlinearity. The Schrödinger 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.
We consider the nonlinear Schrödinger 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.
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.
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 $t\rightarrow0^-$.
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.
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 $\partial_t-Δ_H$, where $Δ_H$ 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.
We consider the scattering problem for the nonlinear Schrödinger 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 $\left(-Δ_{V}\right)^{\frac{s}{2}}$ and $\left(-Δ\right)^{\frac{s}{2}}$ 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 Schrödinger operator $-Δ_{V}$. 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 Schrödinger 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.