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.     

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 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.     

Global existence to generalized Boussinesq equation with combined power-type nonlinearities

The Cauchy problem to the generalized Boussinesq equation with combined power-type nonlinearities is studied. Global solvability or finite time blow-up of the solutions with subcritical initial energy is proved by means of the sign preserving property of the Nehari functional. For generalized Lienard (or generalized Bernoulli) nonlinear terms the critical energy constant is explicitly evaluated. A new method, that can be considered as a modification of the potential well method, is developed. The performed numerical experiments support the theoretical results.     

Existence and blow-up of solution of Cauchy problem for the generalized damped multidimensional Boussinesq equation

We consider the existence, both locally and globally in time, and the blow-up of solutions for the Cauchy problem of the generalized damped multidimensional Boussinesq equation.     

Cauchy problem of generalized Boussinesq equation with combined power‐type nonlinearities

In this paper, we study the Cauchy problem of generalized Boussinesq equation with combined power‐type nonlinearities u_tt ?u_xx + u_xxxx + f(u)_xx = 0, where or . The arguments powered by potential well method combined with some other analysis skills allow us to give the sharp conditions of global well‐posedness. And we also characterize the blow‐up phenomenon. Copyright © 2011 John Wiley & Sons, Ltd.     

On the Cauchy problem for the noncompact Landau-Lifshitz-Gilbert equation

Local well-posedness of the Cauchy problem for the noncompact Landau-Lifshitz-Gilbert equation is investigated via the pseudo-stereographic projection. Existence of global solutions is established for small initial data. In the case of one space dimension global existence theorems are proved for large initial data.     

The Cauchy problem for the generalized Boltzmann equation with dissipative collisions

We prove the global existence, uniqueness, and positivity of solutions to the Cauchy problem, with general initial data, for a class of generalized Boltzmann models with dissipative collisions.     

Local and global existence of solutions for the Cauchy problem of a generalized Boussinesq equation

We consider the local and global existence of solutions for a generalized Boussinesq equation utt−uxx+uxxxx+(uk+1)_xx=0, k>4, with initial data in some homogenous Besov-type space.     

Potential well method for Cauchy problem of generalized double dispersion equations

In this paper we study Cauchy problem of generalized double dispersion equations utt−uxx−uxxtt+uxxxx=f(u)_xx, where f(u)=p|u|, p>1 or u2k, . By introducing a family of potential wells we not only get a threshold result of global existence and nonexistence of solutions, but also obtain the invariance of some sets and vacuum isolating of solutions. In addition, the global existence and finite time blow up of solutions for problem with critical initial conditions E(0)=d, I(u₀)?0 or I(u₀)<0 are proved.     

Initial boundary value problem for generalized logarithmic improved Boussinesq equation

In this paper, we consider the initial boundary value problem for generalized logarithmic improved Boussinesq equation. By using the Galerkin method, logarithmic Sobolev inequality, logarithmic Gronwall inequality, and compactness theorem, we show the existence of global weak solution to the problem. By potential well theory, we show the norm of the solution will grow up as an exponential function as time goes to infinity under some suitable conditions. Furthermore, for the generalized logarithmic improved Boussinesq equation with damped term, we obtain the decay estimate of the energy. Copyright © 2016 John Wiley & Sons, Ltd.     

具阻尼的高维广义Boussinesq方程的Cauchy问题的整体适定性

研究了一类具阻尼的高维广义Boussinesq方程u_(tt)-△u-△u_(tt)+△~2u-k△u_t=△f(u)的Cauchy问题.在没有建立问题局部解存在性理论的情况下,利用位势井方法分析了阻尼系数k与初值及井深之间的关系,得到了整体解存在与不存在的门槛结果.     

Global existence of solution of Cauchy problem for nonlinear pseudo-parabolic equation

In this paper, we prove that the Cauchy problem for the nonlinear pseudo-parabolic equation

vt−αvxxt−βvxx+γvx+fx(v)=φx(vx)+g(v)−αg(v)_xx     

Stability problem of Ulam for generalized forms of Cauchy functional equation

Let G₁ be a vector space and G₂ a Banach space. In this paper, we solve the generalized Hyers-Ulam-Rassias stability problem for a generalized form

     

On the Cauchy problem for a semilinear fractional elliptic equation

We study, for the first time in the literature on the subject, the Cauchy problem for a semilinear fractional elliptic equation. Under an a priori assumption on the solution, we propose the Fourier truncation method for stabilizing the ill-posed problem. A stability estimate of logarithmic type is established.     

Two regularization methods for a Cauchy problem for the Laplace equation

A Cauchy problem for the Laplace equation in a rectangle is considered. Cauchy data are given for y=0, and boundary data are for x=0 and x=π. The solution for 0<y?1 is sought. We propose two different regularization methods on the ill-posed problem based on separation of variables. Both methods are applied to formulate regularized solutions which are stably convergent to the exact one with explicit error estimates.