共查询到20条相似文献,搜索用时 15 毫秒
1.
In this paper we consider the Cauchy problem of multidimensional generalized double dispersion equations utt−Δu−Δutt+Δ2u=Δf(u), where f(u)=ap|u|. By potential well method we prove the existence and nonexistence of global weak solution without establishing the local existence theory. And we derive some sharp conditions for global existence and lack of global existence solution. 相似文献
2.
Qun Lin Yong Hong Wu Ryan Loxton 《Journal of Mathematical Analysis and Applications》2009,353(1):186-4258
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. 相似文献
3.
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. 相似文献
4.
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. 相似文献
5.
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. 相似文献
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7.
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. 相似文献
8.
Xu Runzhang 《Mathematical Methods in the Applied Sciences》2011,34(18):2318-2328
In this paper, we study the Cauchy problem of generalized Boussinesq equation with combined power‐type nonlinearities utt ?uxx + uxxxx + 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. 相似文献
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10.
Masayoshi Tsutsumi 《Journal of Mathematical Analysis and Applications》2008,344(1):157-174
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. 相似文献
11.
In this paper we consider the Cauchy problem of two-dimensional generalized Boussinesq-type equation utt−Δu−Δutt+Δ2u+Δf(u)=0. Under the assumption that 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. 相似文献
12.
Global existence to generalized Boussinesq equation with combined power-type nonlinearities 总被引:1,自引:0,他引:1
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. 相似文献
13.
Global existence of solution of Cauchy problem for nonlinear pseudo-parabolic equation 总被引:1,自引:0,他引:1
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 相似文献
14.
In this paper we consider the Cauchy problem for the generalized hyperelasticrod wave equation which includes the Camassa‐Holm equation and the hyperelastic rod wave equation. Firstly, by using the Kato's theory, we prove that the Cauchy problem for the generalized hyperelastic rod wave is locally well‐posed in Sobolev spaces with . Secondly, we give some conservation laws, some useful conclusions and the precise blow‐up scenario and show that the Cauchy problem for the generalized hyperelastic rod wave equation has smooth solutions which blows up in finite time. Thirdly, we give the blow‐up rate of the strong solutions to the Cauchy problem for the generalized hyperelastic rod wave equation. Finally, we give the lower bound of the maximal existence time of the solution and the lower semicontinuity of existence time of solutions to the generalized hyperelastic rod wave equation. 相似文献
15.
This paper concerns the threshold of global existence and finite time blow up of solutions to the time-dependent focusing Gross-Pitaevskii equation describing the Bose-Einstein condensation of trapped dipolar quantum gases. Via a construction of new cross-constrained invariant sets, it is shown that either the corresponding solution globally exists or blows up in finite time according to some appropriate assumptions about the initial datum. 相似文献
16.
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. 相似文献
17.
Yacheng Liu 《Journal of Mathematical Analysis and Applications》2008,338(2):1169-1187
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. 相似文献
18.
Zhi Qian Chu-Li Fu Zhen-Ping Li 《Journal of Mathematical Analysis and Applications》2008,338(1):479-489
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. 相似文献
19.
J. Ignacio Tello 《Journal of Mathematical Analysis and Applications》2006,324(1):381-396
We consider the Cauchy problem
20.
This paper is devoted to studying the initial‐value problem of the Kawahara equation. By establishing some crucial bilinear estimates related to the Bourgain spaces Xs, b(R2) introduced by Bourgain and homogeneous Bourgain spaces, which is defined in this paper and using I‐method as well as L2 conservation law, we show that this fifth‐order shallow water wave equation is globally well‐posed for the initial data in the Sobolev spaces Hs(R) with $s{>}-\frac{63}{58}$. Copyright © 2010 John Wiley & Sons, Ltd. 相似文献