Periodic solutions of nonlinear wave equations with damping

A known existence theorem for doubly periodic solutions of nonlinear wave equations with linear damping is being proved in a direct manner by an approach which has been developed by the authors in [5, 6, 7] for hyperbolic problems, when the kernel of the underlying linear operator is infinite dimensional.     

On nonlinear wave equations with degenerate damping and source terms

In this article we focus on the global well-posedness of the differential equation , where is a sub-differential of a continuous convex function . Under some conditions on and the parameters in the equations, we obtain several results on the existence of global solutions, uniqueness, nonexistence and propagation of regularity. Under nominal assumptions on the parameters we establish the existence of global generalized solutions. With further restrictions on the parameters we prove the existence and uniqueness of a global weak solution. In addition, we obtain a result on the nonexistence of global weak solutions to the equation whenever the exponent is greater than the critical value , and the initial energy is negative. We also address the issue of propagation of regularity. Specifically, under some restriction on the parameters, we prove that solutions that correspond to any regular initial data such that , are indeed strong solutions.

     

Blow-up for coupled nonlinear wave equations with damping and source

We consider the system of nonlinear wave equations $\{\begin{array}{cc}{u}_{tt}+u_t+{\left|u_t\right|}^{m?1}{u}_t=\mathrm{div}\left({\rho }_1\left({\left|?u\right|}^2\right)?u\right)+f_1(u,v)\text{,}\hfill & (x,t)\in \Omega \times (0,T)\text{,}\hfill \\ v_{tt}+v_t+{\left|v_t\right|}^{r?1}{v}_t=\mathrm{div}\left({\rho }_2\left({\left|?v\right|}^2\right)?v\right)+f_2(u,v)\text{,}\hfill & (x,t)\in \Omega \times (0,T)\text{,}\hfill \end{array}$ with initial and Dirichlet boundary conditions. Under some suitable assumptions on the functions$f_1$, $f_2$, ${\rho }_1$, ${\rho }_2$, parameters $r,m$ and the initial data, the result on blow-up of solutions and upper bound of blow-up time are given.     

On solutions of quasilinear wave equations with nonlinear damping terms

In this paper we consider the existence and asymptotic behavior of solutions of the following problem:

Cauchy problem for quasi-linear wave equations with nonlinear damping and source terms

The paper studies the existence and non-existence of global weak solutions to the Cauchy problem for a class of quasi-linear wave equations with nonlinear damping and source terms. It proves that when α?max{m,p}, 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, under rather mild conditions on initial data, the Cauchy problem admits a global weak solution. Especially in the case of space dimension N=1, the weak solutions are regularized and so generalized and classical solution both prove to be unique. On the other hand, if 0?α<1, and the initial energy is negative, then under certain opposite conditions, any weak solution of the Cauchy problem blows up in finite time. And an example is shown.     

Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping

We consider the wave equation with supercritical interior and boundary sources and damping terms. The main result of the paper is local Hadamard well-posedness of finite energy (weak) solutions. The results obtained: (1) extend the existence results previously obtained in the literature (by allowing more singular sources); (2) show that the corresponding solutions satisfy Hadamard well-posedness conditions during the time of existence. This result provides a positive answer to an open question in the area and it allows for the construction of a strongly continuous semigroup representing the dynamics governed by the wave equation with supercritical sources and damping.     

Global attractors for wave equations with nonlinear interior damping and critical exponents

In this paper we study the global attractors for wave equations with nonlinear interior damping. We prove the existence, regularity and finite dimensionality of the global attractors without assuming a large value for the damping parameter, when the growth of the nonlinear terms is critical.     

Global Cauchy problem for wave equations with a nonlinear damping term

We study the global Cauchy problem for wave equations with a nonlinear damping term. The existence, uniqueness and nonuniqueness, and nonexistence of a global classical solution are considered.     

Global existence and nonexistence for nonlinear wave equations with damping and source terms

We consider an initial-boundary value problem for a nonlinear wave equation in one space dimension. The nonlinearity features the damping term and a source term of the form , with 1$">. We show that whenever , then local weak solutions are global. On the other hand, we prove that whenever m$"> and the initial energy is negative, then local weak solutions cannot be global, regardless of the size of the initial data.

     

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.     

Solutions in Bessel‐potential spaces for wave equations with nonlinear damping

We consider a semilinear wave equation with nonlinear damping in the whole space . Local‐in‐time existence and uniqueness results are obtained in the class of Bessel‐potential spaces . Copyright © 2017 John Wiley & Sons, Ltd.     

Periodic solutions for wave equations with variable coefficients with nonlinear localized damping

We discuss the existence of periodic solutions to the wave equation with variable coefficients utt−div(A(x)∇u)+ρ(x,ut)=f(x,t) with Dirichlet boundary condition. Here ρ(x,v) is a function like ρ(x,v)=a(x)g(v) with g^′(v)?0 where a(x) is nonnegative, being positive only in a neighborhood of a part of the domain.     

The lifespan for 3D quasilinear wave equations with nonlinear damping terms

In this paper, we study the initial-boundary value problem for a system of nonlinear wave equations, involving nonlinear damping terms, in a bounded domain Ω. The nonexistence of global solutions is discussed under some conditions on the given parameters. Estimates on the lifespan of solutions are also given. Our results extend and generalize the recent results in [K. Agre, M.A. Rammaha, System of nonlinear wave equations with damping and source terms, Differential Integral Equations 19 (2006) 1235-1270], especially, the blow-up of weak solutions in the case of non-negative energy.     

Symplectic integration of Hamiltonian wave equations

Summary The numerical integration of a wide class of Hamiltonian partial differential equations by standard symplectic schemes is discussed, with a consistent, Hamiltonian approach. We discretize the Hamiltonian and the Poisson structure separately, then form the the resulting ODE's. The stability, accuracy, and dispersion of different explicit splitting methods are analyzed, and we give the circumstances under which the best results can be obtained; in particular, when the Hamiltonian can be split into linear and nonlinear terms. Many different treatments and examples are compared.     

Stochastic wave equations with dissipative damping

We prove existence and (in some special case) uniqueness of an invariant measure for the transition semigroup associated with the stochastic wave equations with nonlinear dissipative damping.     

Resonances and O-curves in Hamiltonian systems

We investigate the problem of the existence of trajectories asymptotic to elliptic equilibria of Hamiltonian systems in the presence of resonances.