Shihui Zhu

• Sichuan Normal University

In this paper, we study the Cauchy problem for focusing nonlinear beam equations with and without a damping term. By constructing two pairs of invariant flows, we obtain the exact thresholds for the global existence and blow-up to the above equations in the sense that both thresholds are explicitly expressed by the L ² -norm of the fourth-order non...

In this paper, we study the ground state standing wave solutions for the focusing bi-harmonic nonlinear Schrödinger equation with a μ-Laplacian term (BNLS). Such BNLS models the propagation of intense laser beams in a bulk medium with a second-order dispersion term. Denoting by Qp the ground state for the BNLS with μ = 0, we prove that in the mass-...

By introducing and solving a new cross-constrained variational problem, a one-to-one correspondence from the prescribed mass to frequency of soliton is established for the generalized Davey-Stewartson system in two-dimensional space. Orbital stability of small soiltons depending on frequencies is proved. Multisolitons with different speeds are cons...

In this paper, we identify the sharp energy threshold for the global existence and blow-up to the Hartree equation with a focusing and defocusing perturbation, respectively, by utilizing comprehensive potential-well structures generated by the combined nonlinearities. Then, by adopting the refined compactness lemma, we detect the dynamical properti...

This study investigates the two-dimensional (2D) Gross–Pitaevskii–Poisson equations that model dipolar Bose–Einstein condensation. The new element introduced to these equations in this study is the construction of the modified cross-constrained minimization problem dM~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepacka...

This paper is concerned with the generalized Davey-Stewarston system in two dimensional space. Existence and stability of small solitons are proved by solving two correlative constrained variational problems and spectrum analysis. In addition, multi-solitons with different speeds are constructed by bootstrap argument.

In this paper, we make a comprehensive study for the orbital stability of standing waves for the fractional Schrödinger equation with combined power-type nonlinearities(FNLS)i∂tψ−(−Δ)sψ+a|ψ|p1ψ+|ψ|p2ψ=0. We prove that when p2=4sN and a(p1−4sN)<0, there exist the standing waves of (FNLS), which are orbitally stable. When a=0 and 4sN<p2<4sN−2s, we pr...

This paper studies the blow‐up solutions for the Schrödinger equation with a Hartree‐type nonlinearity together with a power‐type subcritical perturbation. The precisely sharp energy thresholds for blow‐up and global existence are obtained by analyzing potential well structures for associated functionals.

The fourth-order complete discrimination system method is employed to the seek exact solutions of the Tzitzeica–Dodd–Bullough equation as well as the double sinh-Gordon equation. We obtain a range of new solutions, including a periodic solution, a triangular function solution, rational function solutions, peakon solutions and Jacobian elliptic func...

We prove the existence of the set of ground states in a suitable energy space $\Sigma^s=\{u: \int_{\mathbb{R}^N} \bar{u}(-\Delta+m^2)^s u+V |u|^2<\infty\}$, $s\in (0,\frac{N}{2})$ for the mass-subcritical nonlinear fractional Hartree equation with unbounded potentials. As a consequence we obtain, as a priori result, the orbital stability of the set...

This paper investigates dipolar Bose‐Einstein condensate modeled by the Gross‐Pitaevskii equation with harmonic confinement and dipolar interaction potential. With the help of a new estimate of the kinetic energy as well as a mechanical analogy, the known results of collapse[14] have been extended to a new ample condition for the existence of blowu...

In this paper, we study the standing wave solutions of the bi-harmonic nonlinear Schr\"{o}dinger equation with the Laplacian term (BNLS). By taking into account the role of second-order dispersion term in the propagation of intense laser beams in a bulk medium with Kerr nonlinearity, we prove that in the mass-subcritical regime $p\in (1,1+\frac{8}{...

We consider the focusing nonlinear Schrödinger equation with inverse square potential i∂tu + ∆u + c|x| −2 u = −|u| α u, u(0) = u 0 ∈ H 1 , (t, x) ∈ R + × R d , where d ≥ 3, c = 0, c < λ(d) = d−2 2 2 and 0 < α ≤ 4 d. Using the profile decomposition obtained recently by the first author [1], we show that in the L 2-subcritical case, i.e. 0 < α < 4 d...

The current paper contemplates the blowup dynamics in trapped dipolar quantum gases. More precisely, employing the profile decomposition of bounded sequences in H˙¹∩H˙[Formula presented], we firstly construct related variational problems and derive two refined Gagliardo–Nirenberg inequalities. Secondly, a compactness lemma is utilized to prove that...

In the first part of this paper, we investigate the sharp threshold of blow-up and global existence for the focusing nonlinear Schr\"{o}dinger equation with combined nonlinearities of mass-critical and mass-subcritical power-type. Especially, we find a sequence of initial data with mass approximating that of the ground state from above, the corresp...

In this paper, we establish a triple-order complete discrimination system to derive the traveling wave solutions of the generalized KdV equation with high power nonlinearities, which consist of solitary patterns solutions, compactons solutions, periodic solutions and Jacobi elliptic functions solutions.

We prove that there exist stable standing waves for the fractional Schrödinger equation with combined nonlinearities, one of which is the critical nonlinearity, and these standing waves are orbitally stable.

In this paper, the sharp threshold of scattering for the fractional nonlinear Schr\"{o}dinger equation in the $L^2$-supercritical case is obtained, i.e., if $1+\frac{4s}{N}<p<1+\frac{4s}{N-2s}$, and $$ M[u_{0}]^{\frac{s-s_c}{s_c}}E[u_{0}]<M[Q]^{\frac{s-s_c}{s_c}}E[Q], \ M[u_{0}]^{\frac{s-s_c}{s_c}}\| u_{0}\|^2_{\dot H^s}<M[Q]^{\frac{s-s_c}{s_c}}\|...

We consider the fractional Hartree equation in the $L^2$-supercritical case, and we find a sharp threshold of the scattering versus blow-up dichotomy for radial data: If $ M[u_{0}]^{\frac{s-s_c}{s_c}}E[u_{0}<M[Q]^{\frac{s-s_c}{s_c}}E[Q]$ and $M[u_{0}]^{\frac{s-s_c}{s_c}}\|u_{0}\|^2_{\dot H^s}<M[Q]^{\frac{s-s_c}{s_c}}\| Q\|^2_{\dot H^s}$, then the s...

In this paper, we study the global well-posedness for the CamassaHolm(C-H) equation with a forcing in H-1(R) by the characteristic method. Due to the forcing, many important properties to study the well-posedness of weak solutions do not inherit from the C-H equation without a forcing, such as conservation laws, integrability. By exploiting the bal...

This paper is dedicated to the blow-up solutions for the nonlinear fractional Schrödinger equation arising from pseudorelativistic Boson stars. First, we compute the best constant of a gG-N inequality by the profile decomposition theory and variational arguments. Then, we find the sharp threshold mass of the existence of finite-time blow-up solutio...

This paper is concerned with the blow-up solutions for the Davey-Stewartson system with competing nonlinearities, which results in the loss of scaling invariance. The best constant of a new gG-N type inequality is given to find the sharp threshold mass of blow-up and global existence. Moreover, under the sharp threshold mass, the dynamical behavior...

In this paper, we study the global well-posedness for the Camassa-Holm(C-H) equation with a forcing in $H^1(\mathbb{R})$ by the characteristic method. Due to the forcing, many important properties to study the well posedness of weak solutions do not inherit from the C-H equation without a forcing, such as conservation laws, integrability. By exploi...

In this paper, we study the sharp energy criteria of blow-up and global existence for the nonlinear Klein-Gordon equation by the sharp Gagliardo-Nirebergy-Sobolev inequality. MSC: 35Q40, 35L05.

We study the standing waves of the nonlinear fractional Schrödinger equation. We obtain that when (Formula presented.), the standing waves are orbitally stable; when (Formula presented.), the ground state solitary waves are strongly unstable to blow-up.

In this paper, we study the dynamics of blow-up solutions for the nonlinear inhomogeneous Schrodinger equation. Firstly, we show the lower blow-up rate of blow-up solutions by rescaling technique, and use it to get the rate of mass concentration of blow-up solutions. Secondly, for the minimal mass blow-up solutions, we obtain the sharp lower blow-u...

In this paper, using the variational characteristic of the virial identity and a new estimate of the kinetic energy, we obtain a new sufficient condition for the existence of blow-up solutions. MSC: 35Q55, 35B44.

This paper studies blow-up solutions for the inhomogeneous Schrodinger equation with L-2 supercritical nonlinearity. In terms of Strauss' arguments in Strauss (1977) [22], we find a new compactness lemma for radial symmetric functions. Thus, we use it to derive the best constants of two generalized Gagliardo-Nirenberg type inequalities. Moreover, w...

The global well-posedness of rough solutions to the Cauchy problem for the Davey-Stewartson system is obtained. It reads that if the initial data is in with s > 2/5, then there exists a global solution in time, and the norm of the solution obeys polynomial-in-time bounds. The new ingredient in this paper is an interaction Morawetz estimate, which g...

This paper is concerned with the blow-up solutions of nonlinear Schrödinger equation (NLS) with oscillating nonlinearities. The limiting profiles of the blow-up solutions u(t, x) with initial data {pipe}{pipe}u 0{pipe}{pipe}L 2={pipe}{pipe}Q{pipe}{pipe}L 2 are obtained. It reads that {pipe}u(t,x){pipe} 2 (Dirac function), as t→T, and that u(t, x) c...

This article studies the Cauchy problem for the damped wave equation with nonlinear memory. For a noncompactly supported initial data with small energy, global existence and asymptotic behaviour of solutions are obtained when 1 ≤ n ≤ 3. This result generalized the previous result by Fino [Critical exponent for damped wave equations with nonlinear m...

This paper deals with the formation of singularities of rough blow-up solutions to the fourth-order nonlinear Schrödinger equation. The limiting profile and L2L2-concentration of the rough blow-up solutions are obtained in Hs(R4)Hs(R4) with s>s0s>s0, where s0≤9+72120≈1.793. The new ingredient relies on the refined compactness result developed by Zh...

This paper is concerned with the Cauchy problem for the biharmonic nonlinear Schrödinger equation with L2L2-super-critical nonlinearity. By establishing the profile decomposition of bounded sequences in H2(RN)H2(RN), the best constant of a Gagliardo–Nirenberg inequality is obtained. Moreover, a sufficient condition for the global existence of the s...

We study the blow-up solutions for the Davey-Stewartson system iu t +Δu+|u| p-1 u+E(|u| 2 )u=0,t≥0,x∈ℝ 3 ·(1) Using the profile decomposition of the bounded sequences in H 1 (ℝ 3 ), we give some new variational characteristics for the ground states and generalized Gagliardo-Nirenberg inequalities. Then, we obtain the precise expressions on the shar...

We consider the blow-up solutions of the Cauchy problem for the critical nonlinear Schrödinger equation with a repulsive harmonic potential. In terms of Merle and Tsutsumi’s arguments as well as Carles’ transform, the L 2 -concentration property of radially symmetric blow-up solutions is obtained.

This article is concerned with the blow-up solutions of the biharmonic Schrödinger equation with L 2-super critical nonlinearity. We obtain the nonexistence of strong limit of L p c -norm and L p c -concentration properties of the radially symmetric blow-up solutions, where L p c is the invariant Lebesgue space.

This paper is concerned with the blow-up solutions of Gross–Pitaevskii equation. We obtain the upper bound of weak-limitation for the blow-up solutions by using the method of Cazenave (2003) [3] as well as the concentration compact principle.

This paper is concerned with the blow-up solutions of the focusing fourth-order mass-critical nonlinear Schrödinger equation. Establishing the profile decomposition of the bounded sequences in H 2 , we obtain the variational characteristics of the corresponding ground state and a compactness lemma. Moreover, we obtain the L 2 -concentration of the...

This paper is concerned with the blow-up solutions of the Gross-Pitaevskii equation. Using the concentration compact principle and the variational characterization of the corresponding ground state, we obtain the limiting profile of blow-up solutions with critical mass in the corresponding weighted energy space. Moreover, we extend this result to s...

We consider the blow-up solutions of the Cauchy problem for the critical nonlinear Schrödinger equation with Stark potential and the sharp lower and upper bounds of blow-up rate are established.

We consider the blow-up solutions of the Cauchy problem for critical nonlinear Schrödinger equation with a repulsive harmonic potential. In terms of Merle and Raphaël’s recent arguments as well as Carles’ transform, the sharp upper and lower bounds of the blow-up rate are obtained.