Blow-up result and energy decay rates for binary mixtures of solids with nonlinear damping and source terms

In this paper we study the long-time behavior of binary mixture problem of solids, focusing on the interplay between nonlinear damping and source terms. 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 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.     

Long-time behavior of solutions to a nonlinear hyperbolic relaxation system

We study the large time behavior of solutions of a one-dimensional hyperbolic relaxation system that may be written as a nonlinear damped wave equation. First, we prove the global existence of a unique solution and their decay properties for sufficiently small initial data. We also show that for some large initial data, solutions blow-up in finite time. For quadratic nonlinearities, we prove that the large time behavior of solutions is given by the fundamental solution of the viscous Burgers equation. In some other cases, the convection term is too weak and the large time behavior is given by the linear heat kernel.     

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.     

Global existence, asymptotic behavior and blow-up of solutions for coupled Klein-Gordon equations with damping terms

This paper studies the Cauchy problem for the coupled system of nonlinear Klein-Gordon equations with damping terms. We first state the existence of standing wave with ground state, based on which we prove a sharp criteria for global existence and blow-up of solutions when E(0)<d. We then introduce a family of potential wells and discuss the invariant sets and vacuum isolating behavior of solutions for 0<E(0)<d and E(0)≤0, respectively. Furthermore, we prove the global existence and asymptotic behavior of solutions for the case of potential well family with 0<E(0)<d. Finally, a blow-up result for solutions with arbitrarily positive initial energy is obtained.     

Global well-posedness and stability of semilinear Mindlin–Timoshenko system

We study long-term behavior of Reissner–Mindlin–Timoshenko (RMT) plate systems, focusing on the interplay between nonlinear viscous 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. This work complements [28] which established local well-posedness of this problem, global well-posedness when damping dominates the sources (in an appropriate sense) and a blow-up in the complementary scenario assuming negative “total” initial energy. The current paper develops the potential well theory for the RMT system: it proves global existence for potential well solutions without restricting the source exponents, derives explicit energy decay rates dependent on the order of the damping exponents, and verifies a blow-up result for positive total initial energy.     

Beam evolution equation with variable coefficients

We investigate a initial‐boundary value problem for the nonlinear beam equation with variable coefficients on the action of a linear internal damping. We show the existence of a unique global weak solution and that the energy associated with this solution has a rate decay estimate. Besides, we prove the existence and uniqueness of non‐local strong solutions. Copyright © 2004 John Wiley & Sons, Ltd.     

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.     

Global existence and compact attractors for the discrete nonlinear Schrödinger equation

We study the asymptotic behavior of solutions of discrete nonlinear Schrödinger-type (DNLS) equations. For a conservative system, we consider the global in time solvability and the question of existence of standing wave solutions. Similarities and differences with the continuous counterpart (NLS-partial differential equation) are pointed out. For a dissipative system we prove existence of a global attractor and its stability under finite-dimensional approximations. Similar questions are treated in a weighted phase space. Finally, we propose possible extensions for various types of DNLS equations.     

Hadamard well-posedness for a hyperbolic equation of viscoelasticity with supercritical sources and damping

Presented here is a study of a viscoelastic wave equation with supercritical source and damping terms. We employ the theory of monotone operators and nonlinear semigroups, combined with energy methods to establish the existence of a unique local weak solution. In addition, it is shown that the solution depends continuously on the initial data and is global provided the damping dominates the source in an appropriate sense.     

Global well-posedness for strongly damped viscoelastic wave equation

We study the initial boundary value problem for the nonlinear viscoelastic wave equation with strong damping term and dispersive term. By introducing a family of potential wells we not only obtain the invariant sets, but also prove the existence and nonexistence of global weak solution under some conditions with low initial energy. Furthermore, we establish a blow-up result for certain solutions with arbitrary positive initial energy (high energy case)     

Existence,Uniqueness and Asymptotic Behavior for the Vlasov–Poisson System with Radiation Damping

We investigate the Cauchy problem for the Vlasov–Poisson system with radiation damping.By virtue of energy estimate and a refined velocity average lemma, we establish the global existence of nonnegative weak solution and asymptotic behavior under the condition that initial data have finite mass and energy. Furthermore, by building a Gronwall inequality about the distance between the Lagrangian flows associated to the weak solutions, we can prove the uniqueness of weak solution when the initial data have a higher order velocity moment.     

Global existence, asymptotic behavior and blowup of solutions for a class of nonlinear wave equations with dissipative term

The paper studies the global existence, asymptotic behavior and blowup of solutions to the initial boundary value problem for a class of nonlinear wave equations with dissipative term. It proves that under rather mild conditions on nonlinear terms and initial data the above-mentioned problem admits a global weak solution and the solution decays exponentially to zero as t→+∞, respectively, in the states of large initial data and small initial energy. In particular, in the case of space dimension N=1, the weak solution is regularized to be a unique generalized solution. And if the conditions guaranteeing the global existence of weak solutions are not valid, then under the opposite conditions, the solutions of above-mentioned problem blow up in finite time. And an example is given.     

On a quasilinear parabolic–elliptic chemotaxis system with logistic source

This paper deals with a quasilinear parabolic–elliptic chemotaxis system with logistic source, under homogeneous Neumann boundary conditions in a smooth bounded domain. For the case of positive diffusion function, it is shown that the corresponding initial boundary value problem possesses a unique global classical solution which is uniformly bounded. Moreover, if the diffusion function is zero at some point, or a positive diffusion function and the logistic damping effect is rather mild, we proved that the weak solutions are global existence. Finally, it is asserted that the solutions approach constant equilibria in the large time for a specific case of the logistic source.     

On a wave equation with supercritical interior and boundary sources and damping terms

This article addresses nonlinear wave equations with supercritical interior and boundary sources, and subject to interior and boundary damping. The presence of a nonlinear boundary source alone is known to pose a significant difficulty since the linear Neumann problem for the wave equation is not, in general, well‐posed in the finite‐energy space H¹(Ω) × L₂(?Ω) with boundary data in L₂ due to the failure of the uniform Lopatinskii condition. Further challenges stem from the fact that both sources are non‐dissipative and are not locally Lipschitz operators from H¹(Ω) into L₂(Ω), or L₂(?Ω). With some restrictions on the parameters in the model and with careful analysis involving the Nehari Manifold, we obtain global existence of a unique weak solution, and establish exponential and algebraic uniform decay rates of the finite energy (depending on the behavior of the dissipation terms). Moreover, we prove a blow up result for weak solutions with nonnegative initial energy.     

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.     

Strong solutions for two-dimensional nonlocal Cahn–Hilliard–Navier–Stokes systems

A well-known diffuse interface model for incompressible isothermal mixtures of two immiscible fluids consists of the Navier–Stokes system coupled with a convective Cahn–Hilliard equation. In some recent contributions the standard Cahn–Hilliard equation has been replaced by its nonlocal version. The corresponding system is physically more relevant and mathematically more challenging. Indeed, the only known results are essentially the existence of a global weak solution and the existence of a suitable notion of global attractor for the corresponding dynamical system defined without uniqueness. In fact, even in the two-dimensional case, uniqueness of weak solutions is still an open problem. Here we take a step forward in the case of regular potentials. First we prove the existence of a (unique) strong solution in two dimensions. Then we show that any weak solution regularizes in finite time uniformly with respect to bounded sets of initial data. This result allows us to deduce that the global attractor is the union of all the bounded complete trajectories which are strong solutions. We also demonstrate that each trajectory converges to a single equilibrium, provided that the potential is real analytic and the external forces vanish.     

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.     

Cauchy problem for quasi-linear wave equations with viscous damping

The paper studies the global existence and asymptotic behavior of weak solutions to the Cauchy problem for quasi-linear wave equations with viscous damping. It proves that when pmax{m,α}, where m+1, α+1 and p+1 are, respectively, the growth orders of the nonlinear strain terms, the nonlinear damping term and the source term, the Cauchy problem admits a global weak solution, which decays to zero according to the rate of polynomial as t→∞, as long as the initial data are taken in a certain potential well and the initial energy satisfies a bounded condition. Especially in the case of space dimension N=1, the solutions are regularized and so generalized and classical solution both prove to be unique. Comparison of the results with previous ones shows that there exist clear boundaries similar to thresholds among the growth orders of the nonlinear terms, the states of the initial energy and the existence, asymptotic behavior and nonexistence of global solutions of the Cauchy problem.     

A unified approach via convexity for optimal energy decay rates of finite and infinite dimensional vibrating damped systems with applications to semi-discretized vibrating damped systems

The Liapunov method is celebrated for its strength to establish strong decay of solutions of damped equations. Extensions to infinite dimensional settings have been studied by several authors (see e.g. Haraux, 1991 [11], and Komornik and Zuazua, 1990 [17] and references therein). Results on optimal energy decay rates under general conditions of the feedback is far from being complete. The purpose of this paper is to show that general dissipative vibrating systems have structural properties due to dissipation. We present a general approach based on convexity arguments to establish sharp optimal or quasi-optimal upper energy decay rates for these systems, and on comparison principles based on the dissipation property, and interpolation inequalities (in the infinite dimensional case) for lower bounds of the energy. We stress the fact that this method works for finite as well as infinite dimensional vibrating systems and as well as for applications to semi-discretized nonlinear damped vibrating PDE's. A part of this approach has been introduced in Alabau-Boussouira (2004, 2005) [1] and [2]. In the present paper, we identify a new, simple and explicit criteria to select a class of nonlinear feedbacks, for which we prove a simplified explicit energy decay formula comparatively to the more general but also more complex formula we give in Alabau-Boussouira (2004, 2005) [1] and [2]. Moreover, we prove optimality of the decay rates for this class, in the finite dimensional case. This class includes a wide range of feedbacks, ranging from very weak nonlinear dissipation (exponentially decaying in a neighborhood of zero), to polynomial, or polynomial-logarithmic decaying feedbacks at the origin. In the infinite dimensional case, we establish a comparison principle on the energy of sufficiently smooth solutions through the dissipation relation. This principle relies on suitable interpolation inequalities. It allows us to give lower bounds for the energy of smooth initial data for the one-dimensional wave equation with a distributed polynomial damping, which improves Haraux (1995) [12] lower estimate of the energy for this case. We also establish lower bounds in the multi-dimensional case for sufficiently smooth solutions when such solutions exist. We further mention applications of these various results to several classes of PDE's, namely: the locally and boundary damped multi-dimensional wave equation, the locally damped plate equation and the globally damped coupled Timoshenko beams system but it applies to several other examples. Furthermore, we show that these optimal energy decay results apply to finite dimensional systems obtained from spatial discretization of infinite dimensional damped systems. We illustrate these results on the one-dimensional locally damped wave and plate equations discretized by finite differences and give the optimal energy decay rates for these two examples. These optimal rates are not uniform with respect to the discretization parameter. We also discuss and explain why optimality results have to be stated differently for feedbacks close to linear behavior at the origin.     

Global existence and nonexistence for a nonlinear wave equation with damping and source terms

In this paper we consider a nonlinear wave equation with damping and source term on the whole space. For linear damping case, we show that the solution blows up in finite time even for vanishing initial energy. The criteria to guarantee blowup of solutions with positive initial energy are established both for linear and nonlinear damping cases. Global existence and large time behavior also are discussed in this work. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)