Global existence,asymptotic behavior and uniform attractors for non-autonomous thermoelastic systems

In this paper, we first establish the global existence and asymptotic behavior of solutions by using the semigroup method and multiplicative techniques, then further prove the existence of a uniform attractor for a non-autonomous thermoelastic system by using the method of uniform contractive functions. The main advantage of this method is that we need only to verify compactness condition with the same type of energy estimates as that for establishing absorbing sets. Moreover, we also investigate an alternative result of solutions to the semilinear thermoelastic systems by virtue of the semigroup method.     

The global existence of solutions for two classes of chemotaxis models

This paper is concerned with the global existence and uniform boundedness of solutions for two classes of chemotaxis models in two or three dimensional spaces. Firstly, by using detailed energy estimates, special interpolation relation and uniform Gronwall inequality, we prove the global existence of uniformly bounded solutions for a class of chemotaztic systems with linear chemotactic-sensitivity terms and logistic reaction terms. Secondly, by applying detailed analytic semigroup estimates and special iteration techniques, we obtain the global existence of uniformly bounded solutions for a class of chemotactic systems with nonlinear chemotacticsensitivity terms, which extends the global existence results of [6] to other general cases.     

Global existence and general decay for a nonlinear viscoelastic equation with nonlinear localized damping and velocity-dependent material density

In this article, we investigate a nonlinear viscoelastic equation with nonlinear localized damping and velocity-dependent material density. We prove the global existence of weak solutions and general decay of the energy by using the Faedo–Galerkin method [Z.Y. Zhang and X.J. Miao, Global existence and uniform decay for wave equation with dissipative term and boundary damping, Comput. Math. Appl. 59 (2010), pp. 1003–1018; J.Y. Park and J.R. Kang, Global existence and uniform decay for a nonlinear viscoelastic equation with damping, Acta Appl. Math. 110 (2010), pp. 1393–1406] and the perturbed energy method [Zhang and Miao (2010); X.S. Han, and M.X. Wang, Global existence and uniform decay for a nonlinear viscoelastic equation with damping, Nonlinear Anal. TMA. 70 (2009), pp. 3090–3098], respectively. Furthermore, for certain initial data and suitable conditions on the relaxation function, we show that the energy decays exponentially or polynomially depending the rate of the decay of the relaxation function. This result is an improvement over the earlier ones in the literature.     

Global existence,asymptotic stability and blow-up of solutions for the generalized Boussinesq equation with nonlinear boundary condition

In this paper, we consider initial boundary value problem of the generalized Boussinesq equation with nonlinear interior source and boundary absorptive terms. We establish firstly the local existence of solutions by standard Galerkin method. Then we prove both the global existence of the solution and a general decay of the energy functions under some restrictions on the initial data. We also prove a blow-up result for solutions with positive and negative initial energy respectively.     

Existence, blow-up and exponential decay estimates for a nonlinear wave equation with boundary conditions of two-point type

This paper is devoted to studying a nonlinear wave equation with boundary conditions of two-point type. First, we state two local existence theorems and under the suitable conditions, we prove that any weak solutions with negative initial energy will blow up in finite time. Next, we give a sufficient condition to guarantee the global existence and exponential decay of weak solutions. Finally, we present numerical results.     

Uniform bounds from bounded Lp-functionals in reaction-diffusion equations

To prove global existence of classical or mild solutions of reaction-diffusion equations, a priori bounds in the uniform norm are needed. But for interesting examples, often one can only derive bounds for some L_p-norms. Using the structure of the reaction term they can be used to obtain uniform bounds. Two propositions are stated which give conditions for this procedure. The proofs use the smoothing properties of analytic semigroups and the multiplicative Gagliardo-Nirenberg inequality. To illustrate the method, we prove global existence of solutions for the Brusselator and a Volterra-Lotka system with one diffusing and one sedentary species.     

The existence of multiple equilibrium points in a global attractor for some p-Laplacian equation

In this paper, we are mainly concerned with some properties of the global attractor for some p-Laplacian equation with a Lyapunov function in a Banach space. Under some suitable assumptions, we prove the existence of multiple equilibrium points in a global attractor for some p-Laplacian equation.     

The Cahn–Hilliard–Hele–Shaw system with singular potential

The Cahn–Hilliard–Hele–Shaw system is a fundamental diffuse-interface model for an incompressible binary fluid confined in a Hele–Shaw cell. It consists of a convective Cahn–Hilliard equation in which the velocity u is subject to a Korteweg force through Darcy's equation. In this paper, we aim to investigate the system with a physically relevant potential (i.e., of logarithmic type). This choice ensures that the (relative) concentration difference φ takes values within the admissible range. To the best of our knowledge, essentially all the available contributions in the literature are concerned with a regular approximation of the singular potential. Here we first prove the existence of a global weak solution with finite energy that satisfies an energy dissipative property. Then, in dimension two, we further obtain the uniqueness and regularity of global weak solutions. In particular, we show that any two-dimensional weak solution satisfies the so-called strict separation property, namely, if φ is not a pure state at some initial time, then it stays instantaneously away from the pure states. When the spatial dimension is three, we prove the existence of a unique global strong solution, provided that the initial datum is regular enough and sufficiently close to any local minimizer of the free energy. This also yields the local Lyapunov stability of the local minimizer itself. Finally, we prove that under suitable assumptions any global solution converges to a single equilibrium as time goes to infinity.     

Pullback attractors of 2D Navier–Stokes equations with weak damping,distributed delay,and continuous delay

In this present paper, the existence of pullback attractors for the 2D Navier–Stokes equation with weak damping, distributed delay, and continuous delay has been considered, by virtue of classical Galerkin's method, we derived the existence and uniqueness of global weak and strong solutions. Using the Aubin–Lions lemma and some energy estimate in the Banach space with delay, we obtained the uniform bounded and existence of uniform pullback absorbing ball for the solution semi‐processes; we concluded the pullback attractors via verifying the pullback asymptotical compactness by the generalized Arzelà–Ascoli theorem. Copyright © 2016 John Wiley & Sons, Ltd.     

关于衰减Timoshenko系统的解的整体存在性、渐近性及一致吸引子（英文）

In this paper, we consider the Timoshenko system as an initial-boundary value problem in a one-dimensional bounded domain. And then we establish the global existence and asymptotic behavior of solution by using the semigroup method and multiplicative techniques, then further prove the existence of a uniform attractor by using the method of uniform contractive function. The main advantage of this method is that we need only to verify compactness condition with the same type of energy estimate as that for establishing absorbing sets.     

Existence,blow‐up,and exponential decay estimates for a system of semilinear wave equations associated with the helicalflows of Maxwell fluid

The paper is devoted to the study of a system of semilinear wave equations associated with the helical flows of Maxwell fluid. First, based on Faedo–Galerkin method and standard arguments of density corresponding to the regularity of initial conditions, we establish two local existence theorems of weak solutions. Next, we prove that any weak solutions with negative initial energy will blow up in finite time. Finally, we give a sufficient condition to guarantee the global existence and exponential decay of weak solutions via the construction of a suitable Lyapunov functional. Copyright © 2015 John Wiley & Sons, Ltd.     

Timoshenko系统在Gurtin-Pipkin热能定律下解的整体存在性、渐进性及一致吸引子（英文）

In this paper,we first establish the global existence and asymptotic behavior of solutions by using the semigroup method and multiplicative techniques,then further prove the existence of a uniform attractor for Timoshenko systems with Gurtin-Pipkin thermal law by using the method of uniform contractive functions.The main advantages of this method is that we need only to verify compactness condition with the same type of energy estimates as that for establishing absorbing sets.Moreover,we also investigate an alternative result of solutions to the semilinear Timoshenko systems by virtue of the semigroup method.     

On a nonlinear heat equation with viscoelastic term associated with Robin conditions

This paper is devoted to study of a nonlinear heat equation with a viscoelastic term associated with Robin conditions. At first, by the Faedo–Galerkin and the compactness method, we prove existence, uniqueness, and regularity of a weak solution. Next, we prove that any weak solution with negative initial energy will blow up in finite time. Finally, by the construction of a suitable Lyapunov functional, we give a sufficient condition to guarantee the global existence and exponential decay of weak solutions.     

Existence and decay rates for a coupled Balakrishnan-Taylor viscoelastic system with dynamic boundary conditions

In this paper, we are concerned with a coupled viscoelastic wave system with Balakrishnan-Taylor dampings, dynamic boundary conditions, source terms, and past histories. Under suitable assumptions on relaxation functions and source terms, we prove the global existence of solutions by potential well theory and we establish a more general decay result of energy, in which the exponential decay and polynomial decay are only special cases, by introducing suitable energy and perturbed Lyapunov functionals.     

On the convective Cahn-Hilliard equation with degenerate mobility

In this paper, we study the existence of weak solutions for the convective Cahn-Hilliard equation with degenerate mobility. Based on the Schauder type estimates, we establish the global existence of classical solutions for regularized problems. After establishing some necessary uniform estimates on the approximate solutions, we prove the existence of weak solutions. The nonnegativity and the finite speed of propagation of perturbations of solutions are also discussed.     

Global convergence of an isentropic Euler-Poisson system in $\R^+ \times \R^d$

We prove the global-in-time convergence of an Euler-Poisson system near a constant equilibrium state in the whole space $\R^d$, as physical parameters tend to zero. The result follows from the uniform global existence of smooth solutions by means of energy estimates together with compactness arguments. For this purpose, we establish uniform estimates for $\dive \,u$ and $\curle \,u$ instead of $\nabla u$.     

Existence,blow‐up,and exponential decay estimates for a system of nonlinear wave equations with nonlinear boundary conditions

This paper is devoted to the study of a system of nonlinear equations with nonlinear boundary conditions. First, on the basis of the Faedo–Galerkin method and standard arguments of density corresponding to the regularity of initial conditions, we establish two local existence theorems of weak solutions. Next, we prove that any weak solutions with negative initial energy will blow up in finite time. Finally, the exponential decay property of the global solution via the construction of a suitable Lyapunov functional is presented. Copyright © 2013 John Wiley & Sons, Ltd.     

Porous elastic system with nonlinear damping and sources terms

We study the long-time behavior of porous-elastic system, focusing on the interplay between nonlinear damping and source terms. The sources may represent restoring forces, but may also be focusing thus potentially amplifying the total energy which is the primary scenario of interest. By employing nonlinear semigroups and the theory of monotone operators, we obtain several results on the existence of local and global weak solutions, and uniqueness of weak solutions. Moreover, we prove that such unique solutions depend continuously on the initial data. Under some restrictions on the parameters, we also prove that every weak solution to our system blows up in finite time, provided the initial energy is negative and the sources are more dominant than the damping in the system. Additional results are obtained via careful analysis involving the Nehari Manifold. Specifically, we prove the existence of a unique global weak solution with initial data coming from the “good” part of the potential well. For such a global solution, we prove that the total energy of the system decays exponentially or algebraically, depending on the behavior of the dissipation in the system near the origin. We also prove the existence of a global attractor.     

Global existence of solutions for non‐small data to non‐linear spherically symmetric thermoviscoelasticity

We consider some initial–boundary value problems for non‐linear equations of thermoviscoelasticity in the three‐dimensional case. Since, we are interested to prove global existence we consider spherically symmetric problem. We examine the Neumann conditions for the temperature and either the Neumann or the Dirichlet boundary conditions for the elasticity equations. Using the energy method, we are able to obtain some energy estimates in appropriate Sobolev spaces enough to prove existence for all time without any restrictions on data. Due to the spherical symmetricity the constants in the above estimates increase with time so the existence for all finite times is proved only. Copyright © 2003 John Wiley & Sons, Ltd.     

Threshold condition for global existence and blow-up to a radially symmetric drift-diffusion system

For a class of drift-diffusion systems Kurokiba et al. [M. Kurokiba, T. Nagai, T. Ogawa, The uniform boundedness and threshold for the global existence of the radial solution to a drift-diffusion system, Commun. Pure Appl. Anal. 5 (2006) 97-106.] proved global existence and uniform boundedness of the radial solutions when the L₁-norm of the initial data satisfies a threshold condition. We prove in this letter that this result prescribes a region in the plane of masses which is sharp in the sense that if the drift-diffusion system is initiated outside the threshold region of global existence, then blow-up is possible: suitable initial data can be built up in such a way that the corresponding solution blows up in a finite time.