Global existence and blowup of solutions for nonlinear coupled wave equations with viscoelastic terms

In this paper, we study a system of nonlinear coupled wave equations with damping, source, and nonlinear strain terms. We obtain several results concerning local existence, global existence, and finite time blow‐up property with positive initial energy by using Galerkin method and energy method, respectively. Copyright © 2015 John Wiley & Sons, Ltd.     

On decay and blow-up of solutions for a system of nonlinear wave equations

The initial boundary value problem for a system of viscoelastic wave equations of Kirchhoff type with the nonlinear damping and the source terms in a bounded domain is considered. We prove that, under suitable conditions on the nonlinearity of the damping and the source terms and certain initial data in the stable set and for a wider class of relaxation functions, the decay estimates of the energy function is exponential or polynomial depending on the exponents of the damping terms in both equations by using Nakao’s method. Conversely, for certain initial data in the unstable set, we obtain the blow-up of solutions in finite time when the initial energy is nonnegative. This improves earlier results in the literature.     

Global nonexistence for logarithmic wave equations with nonlinear damping and distributed delay terms

This paper is concerned with global nonexistence of solutions for a logarithmic wave equation with nonlinear damping and distributed delay terms. Due to the simultaneous presence of nonlinear damping and logarithmic source terms, we have difficulty in use of the concavity method. Applying the energy estimates, we show the global nonexistence of solutions with not only non-positive initial energy but also positive initial energy.     

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)     

Global existence and decay of solutions of a nonlinear system of wave equations

This work is concerned with a system of two wave equations with nonlinear damping and source terms acting in both equations. Under some restrictions on the nonlinearity of the damping and the source terms, we show that our problem has a unique local solution. Also, we prove that, for some restrictions on the initial data, the rate of decay of the total energy is exponential or polynomial depending on the exponents of the damping terms in both equations.     

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.     

Global solution and asymptotic behavior for the variable coefficient beam equation with nonlinear damping

This paper is concerned with the initial‐boundary value problem for a variable coefficient beam equation with nonlinear damping. Such a model arises from the vertical deflections of a damped extensible elastic inhomogeneous beam whose density depends on time and position. By using the Faedo–Galerkin method and energy method, we obtain the existence and uniqueness of global strong solution. Furthermore, the exponential decay estimate for the total energy is also derived. Copyright © 2015 John Wiley & Sons, Ltd.     

General decay and blowup of solutions for coupled viscoelastic equation of Kirchhoff type with degenerate damping terms

In this work, we consider a nonlinear system of viscoelastic equations of Kirchhoff type with degenerate damping and source terms in a bounded domain. Under suitable assumptions on the initial data, the relaxation functions g_i(i = 1,2) and degenerate damping terms, we obtain global existence of solutions. Then, we prove the general decay result. Finally, we prove the finite time blow‐up result of solutions with negative initial energy. This work generalizes and improves earlier results in the literature.     

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.     

Stable and unstable sets for the Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms

For the Cauchy problem for the nonlinear wave equation with nonlinear damping and source terms we define stable and unstable sets for the initial data. We prove that, if during the evolution the solution enters into the stable set, the solution is global and we are able to estimate the decay rate of the energy. If during the evolution the solution enters into the unstable set, the solution blows up in finite time.     

On a viscoelastic equation with nonlinear boundary damping and source terms: Global existence and decay of the solution

In this paper, we consider a nonlinear viscoelastic wave equation with nonlinear boundary damping and source terms. Under some appropriate assumptions on the relaxation function $g$ and with certain initial data, the global existence of solutions and a general decay for the energy have been established.     

Global existence and stability of a nonlinear wave equation with variable-exponent nonlinearities

ABSTRACT

In this paper, we consider a nonlinear wave equation with damping and source terms of variable-exponent types. First, we use the stable-set method to prove a global result. Then, by applying an integral inequality due to Komornik, we obtain the stability result.     

Cauchy problem for a nonlinear viscoelastic equation with nonlinear damping and source terms

In this paper, we consider a Cauchy problem for a nonlinear viscoelastic equation with nonlinear damping and source terms. Under suitable assumptions on the initial data and the relaxation function, we establish a finite-time blow-up result and a global existence result.     

Uniform decay and blow‐up of solutions for coupled nonlinear Klein–Gordon equations with nonlinear damping terms

In this work, we consider coupled nonlinear Klein–Gordon equations with nonlinear damping terms, in a bounded domain. The decay estimates of the solution are established by using Nakao's inequality. We also prove the blow up of the solution in finite time with negative initial energy. Copyright © 2013 John Wiley & Sons, Ltd.     

Global nonexistence for an integro‐differential equation

The initial boundary value problem for an integro‐differential equation with nonlinear damping and source terms in a bounded domain is considered. By modifying the method in a work by Autuori et al. in 2010, we establish the nonexistence result of global solutions with the initial energy controlled by a critical value. This improves earlier results in the literatures. Copyright © 2011 John Wiley & Sons, Ltd.     

General stability and exponential growth of nonlinear variable coefficient wave equation with logarithmic source and memory term

This paper is concerned with the asymptotic stability and instability of solutions to a variable coefficient logarithmic wave equation with nonlinear damping and memory term. Such model describes wave traveling through nonhomogeneous viscoelastic materials. By choosing appropriate multiplier and using weighted energy method, we prove the exponential decay of the energy. Moreover, we also obtain the instability of the solutions at the infinity in the presence of the nonlinear damping.     

Global nonexistence of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms

This work is concerned with a system of viscoelastic wave equations with nonlinear damping and source terms acting in both equations. We prove a global nonexistence theorem for certain solutions with positive initial energy.     

On decay and blow-up of the solution for a viscoelastic wave equation with boundary damping and source terms

In this paper we consider the decay and blow-up properties of a viscoelastic wave equation with boundary damping and source terms. We first extend the decay result (for the case of linear damping) obtained by Lu et al. (On a viscoelastic equation with nonlinear boundary damping and source terms: Global existence and decay of the solution, Nonlinear Analysis: Real World Applications 12 (1) (2011), 295-303) to the nonlinear damping case under weaker assumption on the relaxation function g(t). Then, we give an exponential decay result without the relation between g^′(t) and g(t) for the linear damping case, provided that ‖g‖_L¹(0,∞) is small enough. Finally, we establish two blow-up results: one is for certain solutions with nonpositive initial energy as well as positive initial energy for both the linear and nonlinear damping cases, the other is for certain solutions with arbitrarily positive initial energy for the linear damping case.     

General decay of solutions of a nonlinear system of viscoelastic wave equations

This work is concerned with a system of two viscoelastic wave equations with nonlinear damping and source terms acting in both equations. Under some restrictions on the nonlinearity of the damping and the source terms, we prove that, for certain class of relaxation functions and for some restrictions on the initial data, the rate of decay of the total energy depends on those of the relaxation functions. This result improves many results in the literature, such as the ones in Messaoudi and Tatar (Appl. Anal. 87(3):247–263, 2008) and Liu (Nonlinear Anal. 71:2257–2267, 2009) in which only the exponential and polynomial decay rates are considered.     

Global existence and blow-up of solutions for some hyperbolic systems with damping and source terms

This paper deals with the global existence and blow-up of solutions to some nonlinear hyperbolic systems with damping and source terms in a bounded domain. By using the potential well method, we obtain the global existence. Moreover, for the problem with linear damping terms, blow-up of solutions is considered and some estimates for the lifespan of solutions are given.