Global existence and bounds for blow‐up time in a class of nonlinear pseudo‐parabolic equations with a memory term

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.     

Global existence and nonexistence for degenerate parabolic systems

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 .

     

A sufficient condition for blowup of solutions to a class of pseudo‐parabolic equations with a nonlocal term

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.     

Global existence and nonexistence for some degenerate and quasilinear parabolic systems

The author discusses the degenerate and quasilinear parabolic system

     

Lifespan for a semilinear pseudo‐parabolic equation

This paper deals with the blow‐up solution to the following semilinear pseudo‐parabolic equation in a bounded domain , which was studied by Luo (Math Method Appl Sci 38(12):2636‐2641, 2015) with the following assumptions on p: and the lifespan for the initial energy J(u₀)<0 is considered. This paper generalizes the above results on the following two aspects:

相似文献   

Global existence and blowup of solutions for a parabolic equation with a gradient term

The author discusses the semilinear parabolic equation with . Under suitable assumptions on and , he proves that, if with , then the solutions are global, while if with 1$">, then the solutions blow up in a finite time, where is a positive solution of , with .

     

Nonexistence results for higher order pseudo‐parabolic equations in the Heisenberg group

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.     

Blow‐up phenomena for a pseudo‐parabolic equation

By the means of a differential inequality technique, we obtain a lower bound for blow‐up time if p and the initial value satisfy some conditions. Also, we establish a blow‐up criterion and an upper bound for blow‐up time under some conditions as well as a nonblow‐up and exponential decay under some other conditions. Copyright © 2014 John Wiley & Sons, Ltd.     

H1‐Galerkin expanded mixed finite element methods for nonlinear pseudo‐parabolic integro‐differential equations

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     

A class of pseudo‐parabolic equations: existence,uniqueness of weak solutions,and error estimates for the Euler‐implicit discretization

In this paper, we investigate a class of pseudo‐parabolic equations. Such equations model two‐phase flow in porous media where dynamic effects are included in the capillary pressure. The existence and uniqueness of a weak solution are proved, and error estimates for an Euler implicit time discretization are obtained. Copyright © 2011 John Wiley & Sons, Ltd.     

一类平均曲率型抛物方程解的存在性和解的熄灭

在Rn有界域上考虑一类带有非线性迁移项的平均曲率型方程div{σ(|　Δu|2)　Δu}+b(u)·　Δu=0的第一类初边值问题.主要得到了弱解的存在性,并且给出了解的熄灭性质及解的L∞估计.     

Global existence and nonexistence of solutions for a viscoelastic wave equation with nonlinear boundary source term

In this paper, we consider the initial boundary value problem for a viscoelastic wave equation with nonlinear boundary source term. First of all, we introduce a family of potential wells and prove the invariance of some sets. Then we establish the existence and nonexistence of global weak solution with small initial energy under suitable assumptions on the relaxation function , nonlinear function , the initial data and the parameters in the equation. Furthermore, we obtain the global existence of weak solution for the problem with critical initial conditions and .     

Global existence and nonexistence of generalized coupled reaction‐diffusion systems

In this paper, we consider the initial boundary value problem for a class of reaction‐diffusion systems with generalized coupled source terms. The assumption on the coupled source terms refers to the single equations and includes many kinds of polynomial growth cases. Under this assumption, the reaction‐diffusion systems have a variational structure, which is the foundation of constructing the potential wells to classify the initial data. In subcritical energy level and critical energy level, which are divided from potential well theory, the global existence solution, blow‐up in finite time solution, and asymptotic behavior of solution are obtained, respectively. Furthermore, we show the sufficient conditions of global well posedness with supercritical energy level by combining with comparison principle and semigroup theory.     

Global existence and nonexistence for a doubly degenerate parabolic system coupled via nonlinear boundary flux

This article deals with the degenerate parabolic system with nonlinear boundary flux. By constructing the self-similar supersolution and subsolution, we obtain the critical global existence curve and the critical Fujita curve for the problem. Especially for the blow-up case, it is rather technical. It comes from the construction of the so-called Zel’dovich-Kompaneetz-Barenblatt profile.     

Some properties of solutions for a fourth‐order parabolic equation

This paper is concerned with a fourth‐order parabolic equation in one spatial dimension. On the basis of Leray–Schauder's fixed point theorem, we prove the existence and uniqueness of global weak solutions. Moreover, we also consider the regularity of solution and the existence of global attractor. Copyright © 2012 John Wiley & Sons, Ltd.     

Blow-up and extinction for a sixth-order parabolic equation

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.     

On the behavior of solutions to the Cauchy problem for a degenerate parabolic equation with inhomogeneous density and a source

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.