Cauchy problem for the multi-dimensional Boussinesq type equation

The paper studies the existence and non-existence of global weak solutions to the Cauchy problem for the multi-dimensional Boussinesq type equation utt−Δu+Δ²u=Δσ(u). It proves that the Cauchy problem admits a global weak solution under the assumptions that σ∈C(R), σ(s) is of polynomial growth order, say p (>1), either , s∈R, where β>0 is a constant, or the initial data belong to a potential well. And the weak solution is regularized and the strong solution is unique when the space dimension N=1. In contrast, any weak solution of the Cauchy problem blows up in finite time under certain conditions. And two examples are shown.     

Well-posedness and scattering of solutions to the sixth-order Boussinesq type equation in the framework of modulation spaces

In this paper, we investigate the initial value problem for the sixth order Boussinesq type equation in the framework of modulation spaces. Under suitable conditions, we first prove that the problem has a unique local solutions and global solutions. Then scattering and stability of solutions are also discussed. The proof is mainly based on the decay properties of the solutions operator in modulation spaces and the contraction mapping principle.     

Local existence of strong solutions to the generalized Boussinesq equations

This paper deals with the local existence and uniqueness of strong solutions for the generalized Boussinesq equations with fractional dissipation. As a corollary, we establish some regularity criteria to guarantee smoothness of solutions.     

Global existence and nonexistence of solution for Cauchy problem of two-dimensional generalized Boussinesq equations

In this paper we consider the Cauchy problem of two-dimensional generalized Boussinesq-type equation _utt−Δu−Δ_utt+Δ²u+Δf(u)=0$u_{tt}-\mathrm{\Delta }u-\mathrm{\Delta }{u}_{tt}+{\mathrm{\Delta }}^{2}u+\mathrm{\Delta }f(u)=0$. Under the assumption that f(u)$f(u)$ is a function with exponential growth at infinity and under some assumptions on the initial data, we prove the existence and nonexistence of global weak solution. There are very few works on Boussinesq equation with nonlinear exponential growth term by potential well theory.     

Cauchy problem for quasi-linear wave equations with viscous damping

The paper studies the global existence and asymptotic behavior of weak solutions to the Cauchy problem for quasi-linear wave equations with viscous damping. It proves that when pmax{m,α}, where m+1, α+1 and p+1 are, respectively, the growth orders of the nonlinear strain terms, the nonlinear damping term and the source term, the Cauchy problem admits a global weak solution, which decays to zero according to the rate of polynomial as t→∞, as long as the initial data are taken in a certain potential well and the initial energy satisfies a bounded condition. Especially in the case of space dimension N=1, the solutions are regularized and so generalized and classical solution both prove to be unique. Comparison of the results with previous ones shows that there exist clear boundaries similar to thresholds among the growth orders of the nonlinear terms, the states of the initial energy and the existence, asymptotic behavior and nonexistence of global solutions of the Cauchy problem.     

Existence and scattering of small solutions to a Boussinesq type equation of sixth order

In this paper, we consider the existence and uniqueness of the global small solution as well as the small data scattering result to the Cauchy problem for a Boussinesq type equation of sixth order with the nonlinear term f(u) behaving as as u→0 in . The main method and techniques used in our paper are the Littlewood-Paley dyadic decomposition, the stationary phase estimate and some properties of Bessel function.     

The Cauchy problem for the dissipative Boussinesq equation

This work is devoted to the solvability and finite time blow-up of solutions of the Cauchy problem for the dissipative Boussinesq equation in all space dimension. We prove the existence and uniqueness of local mild solutions in the phase space by means of the contraction mapping principle. By establishing the time-space estimates of the corresponding Green operators, we obtain the continuation principle. Under some restriction on the initial data, we further study the results on existence and uniqueness of global solutions and finite time blow-up of solutions with the initial energy at three different level. Moreover, the sufficient and necessary conditions of finite time blow-up of solutions are given.     

Existence and asymptotic behavior of radially symmetric ground states of quasi-linear singular elliptic equations

Existence and asymptotic behavior of entire positive solutions of a class of quasi-linear elliptic equation is obtained. Under several hypotheses on the ρ(x) and f(r), we obtain the existence of positive entire solution. Asymptotic behavior is discussed by constructing an upper solution. The results of this paper is new and extend previously known results.     

Existence of multi-valued solutions with asymptotic behavior of Hessian equations

In this paper, we use the Perron method to prove the existence of multi-valued solutions with asymptotic behavior at infinity of Hessian equations.     

On the Cauchy problem for a generalized Boussinesq equation

In this paper, we consider the Cauchy problem for a generalized Boussinesq equation. We show that, under suitable conditions, a global solution for the initial value problem exists. In addition, we derive the sufficient conditions for the blow-up of the solution to the problem.     

Existence and mapping properties of the wave operator for the Schrödinger equation with singular potential

We consider the Schrödinger equation in three-dimensional space with small potential in the Lorentz space and we prove Strichartz-type estimates for the solution to this equation. Moreover, using Cook's method, we prove the existence of the wave operator. In the last section we prove the equivalence between the homogeneous Sobolev spaces and in the case .

     

Blowup of solutions for improved Boussinesq type equation

The paper studies the existence and uniqueness of local solutions and the blowup of solutions to the initial boundary value problem for improved Boussinesq type equation u_tt−u_xx−u_xxtt=σ(u)_xx. By a Galerkin approximation scheme combined with the continuation of solutions step by step and the Fourier transform method, it proves that under rather mild conditions on initial data, the above-mentioned problem admits a unique generalized solution u∈W^2,∞([0,T];H²(0,1)) as long as . In particular, when σ(s)=as^p, where a≠0 is a real number and p>1 is an integer, specially a<0 if p is an odd number, the solution blows up in finite time. Moreover, two examples of blowup are obtained numerically.     

On global well-posedness for the 3D Boussinesq equations with fractional partial dissipation

The present paper is dedicated to the global-in-time existence and uniqueness issue for the three-dimensional incompressible Boussinesq equations with fractional partial dissipation.     

New solitary wave solutions for the bad Boussinesq and good Boussinesq equations

In this article, the sine–cosine, the standard tanh and the extended tanh methods has been used to obtain solutions of the bad Boussinesq and good Boussinesq equations. New solitions and periodic solutions are formally derived. The change of parameters, that will drastically change characteristics of the equation, is examined. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2009     

Global solutions of singular parabolic equations arising from electrostatic MEMS

We study dynamic solutions of the singular parabolic problem

(P)     

Existence Uniqueness and Decay of Solution for Fractional Boussinesq Approximation

The Boussinesq approximation finds more and more frequent use in geological practice. In this paper, the asymptotic behavior of solution for fractional Boussinesq approximation is studied. After obtaining some a priori estimates with the aid of commutator estimate, we apply the Galerkin method to prove the existence of weak solution in the case of periodic domain. Meanwhile, the uniqueness is also obtained. Because the results obtained are independent of domain, the existence and uniqueness of the weak solution for Cauchy problem is also true. Finally, we use the Fourier splitting method to prove the decay of weak solution in three cases respectively.     

一类Boussinesq型方程整体解的存在性和唯一性

证明一类6阶Boussinesq型方程Cauchy问题整体广义解和整体古典解的存在性和唯一性,给出解在有限时刻发生爆破的充分条件.