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1.
    
In this article, we consider the initial boundary value problem for a class of nonlinear pseudo‐parabolic equations with a memory term: Under suitable assumptions, we obtain the local and global existence of the solution by Galerkin method. We prove finite‐time blow‐up of the solution for initial data at arbitrary energy level and obtain upper bounds for blow‐up time by using the concavity method. In addition, by means of differential inequality technique, we obtain a lower bound for blow‐up time of the solution if blow‐up occurs.  相似文献   

2.
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3.
    
Sufficient conditions are obtained for the nonexistence of solutions to the nonlinear higher order pseudo‐parabolic equation where is the Kohn‐Laplace operator on the (2N + 1)‐dimensional Heisenberg group , m≥1,p > 1. Then, this result is extended to the case of a 2 × 2‐system of the same type. Our technique of proof is based on judicious choices of the test functions in the weak formulation of the sought solutions. Copyright © 2016 John Wiley & Sons, Ltd.  相似文献   

4.
利用位势井方法研究在有界区域上具有多个非线性源项的波动方程初边值问题．给出了位势井的结构和位势井深度函数的性质．通过引进位势井族得到了在这些问题的流之下的一些集合不变性以及解的真空隔离,揭示了只要问题的初值属于位势井内或位势井外,则问题在今后所有时间内的解都存在于位势井内或井外,同时存在一个没有解的空间区域．进而给出了解的整体存在和不存在的门槛结果．最后,利用相同的方法讨论了具有临界初始条件的问题．  相似文献   

5.
    
A sufficient condition for blowup of solutions to a class of pseudo‐parabolic equations with a nonlocal term is established in this paper. In virtue of the potential wells method, we first extend the results obtained by Xu and Su in [J. Funct. Anal., 264 (12): 2732‐2763, 2013] to the nonlocal case and describe successfully the behavior of solutions by using the energy functional, Nehari functional, and the ground state energy of the stationary equation. Sequently, we study the boundedness and convergency of any global solution. Finally, we achieve a criterion to guarantee the blowup of solutions without any limit of the initial energy.Copyright © 2015 John Wiley & Sons, Ltd.  相似文献   

6.
Global existence and nonexistence for degenerate parabolic systems   总被引：4，自引：0，他引：4  
The initial-boundary value problems are considered for the strongly coupled degenerate parabolic system


in the cylinder , where is bounded and are positive constants. We are concerned with the global existence and nonexistence of the positive solutions. Denote by the first Dirichlet eigenvalue for the Laplacian on . We prove that there exists a global solution iff .

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7.
    
This paper is concerned with the existence, nonexistence, uniqueness, and non‐uniqueness of solutions for a singular parabolic equation in one dimension case. For different cases of some constant m of the equation, we study the uniqueness of solutions for m > ? 1, the nonexistence of solutions for m ≤ ? p, and the existence and non‐uniqueness of solutions for ? p < m ≤ ? 1. The novelty of this paper lies in the study of nonexistence. Moreover, the other results of this paper extend some recent works. Copyright © 2013 John Wiley & Sons, Ltd.  相似文献   

8.
本文讨论一类非局部抛物边值问题的适定性及其数值解法。论证了解的存在、唯一和对问题数据的连续依赖性，然后构造出问题的一个全离散Ｇａｌｅｒｋｉｎ近拟。并给出近似解的稳定性和收敛性分析。数值结果验证了本文方法的有效性。  相似文献   

9.
The author discusses the degenerate and quasilinear parabolic system
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10.
    
We study the Cauchy problem for a class of strongly damped multidimensional generalized Boussinesq equations utt?Δu?Δutt+Δ2u+Δ2utt?kΔut=Δf(u), where k is a positive constant. Under some assumptions and by using potential well method, we prove the existence and nonexistence of global weak solution without solution without establishing the local existence theory. Copyright © 2016 John Wiley & Sons, Ltd.  相似文献   

11.
Huan Liu 《Applicable analysis》2013,92(13):2378-2399
In this paper, we consider an initial-boundary value problem for a sixth-order parabolic equation. We use the modified method of potential wells to study the relationship which the equation solutions existence, blow-up and the asymptotic behavior with initial conditions.  相似文献   

12.
    
In this article we study the initial boundary value problem for a class of fourth-order nonlinear wave equation with viscous damping term u tt ???αu xxt ?+?u xxxx ?=?f(u x ) x . By argument related to the potential well-convexity method, we prove the global existence and nonexistence of the solution. Further, we give some sharp conditions for global existence and nonexistence of the solution. This generalizes the results obtained in Chen and Lu [G. Chen and B. Lu, The initial-boundary value problems for a class of nonlinear wave equations with damping term, J. Math. Anal. Appl. 351 (2009), pp. 1–15].  相似文献   

13.
    
H1‐Galerkin mixed finite element method combined with expanded mixed element method is discussed for nonlinear pseudo‐parabolic integro‐differential equations. We conduct theoretical analysis to study the existence and uniqueness of numerical solutions to the discrete scheme. A priori error estimates are derived for the unknown function, gradient function, and flux. Numerical example is presented to illustrate the effectiveness of the proposed scheme. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013  相似文献   

14.
    
This paper is concerned with a class of fourth‐order nonlinear difference equations. By using the critical point theory, we establish various sets of sufficient conditions of the nonexistence and existence of solutions for Dirichlet boundary value problems and give some new results. Our results successfully complement the existing ones. Copyright © 2014 John Wiley & Sons, Ltd.  相似文献   

15.
The Cauchy problem for a degenerate parabolic equation with a source and inhomogeneous density of the form
$\rho (x)\frac{{\partial u}}{{\partial t}} = div(u^{m - 1} \left| {Du} \right|^{\lambda - 1} Du) + \rho (x)u^p $
is studied. Time global existence and nonexistence conditions are found for a solution to the Cauchy problem. Exact estimates of the solution are obtained in the case of global solvability.
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16.
    
We prove existence and numerical stability of numerical solutions of three fully discrete interior penalty discontinuous Galerkin methods for solving nonlinear parabolic equations. Under some appropriate regularity conditions, we give the l2(H1) and l∞(L2) error estimates of the fully discrete symmetric interior penalty discontinuous Galerkin–scheme with the implicit θ ‐schemes in time, which include backward Euler and Crank–Nicolson finite difference approximations. Our estimates are optimal with respect to the mesh size h. The theoretical results are confirmed by some numerical experiments. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013  相似文献   

17.
The Cauchy problem for a degenerate parabolic equation with a source and variable coefficient of the form
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18.
    
In this paper, we consider the following class of wave equation involving fractional p-Laplacian with logarithmic nonlinearity u t t + ( − Δ ) p s u = | u | q − 2 u log ( | u | ) in Ω , t > 0 , u = 0 in R N ∖ Ω , t > 0 , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = v 0 ( x ) in Ω , $$begin{equation*} hspace*{4pc}{leftlbrace defeqcellsep{&}begin{array}{llc}u_{tt}+(-Delta )^{s}_{p}u=|u|^{q-2}ulog (|u|) & text{in} & Omega ,;t>0 , [3pt] u =0 & text{in} & mathbb {R}^{N}backslash Omega ,;t > 0, [3pt] u(x,0)=u_{0}(x),,,,,u_{t}(x,0)=v_{0}(x)& text{in} &Omega , end{array} right.} end{equation*}$$ where Ω ⊂ R N ( N ≥ 1 ) $Omega subset mathbb {R}^N , (Nge 1)$ is a bounded domain with Lipschitz boundary, s ∈ ( 0 , 1 ) $sin (0,1)$ , 2 ≤ p < p s ∗ $2le p< p^{*}_{s}$ , and p s ∗ = N p N − s p $p^{*}_{s}=frac{Np}{N-sp}$ is the critical exponent in the Sobolev inequality. First, via the Galerkin approximations, the existence of local solutions are obtained when 1 < q < p s ∗ $1<q<p_{s}^{*}$ . Next, by combining the potential well theory with the Nehari manifold, we establish the existence of global solutions when p < q < p s ∗ $p<q<p_{s}^{*}$ . Then, via the Pohozaev manifold, the existence of global solutions are obtained when 1 < q < p s ∗ $1<q<p_{s}^{*}$ . By virtue of a differential inequality technique, we prove that the local solutions blow-up in finite time with arbitrary negative initial energy and suitable initial values. Moreover, we discuss the asymptotic behavior of solutions as time tends to infinity. Here, we point out that the main difficulty is the lack of logarithmic Sobolev inequality concerning fractional p-Laplacian.  相似文献   

19.
    
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20.
In this paper,we consider the linearly viscoelastic equations for Koiter shells. Also,we prove the existence and uniqueness of the solution by Galerkin method.  相似文献   

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