A blow up result for a fractionally damped wave equation

In this paper we prove a blow up result for solutions of the wave equation with damping of fractional order and in presence of a polynomial source. This result improves a previous result in [5]. There we showed that the classical energy is unbounded provided that the initial data are large enough.     

Almost global existence for nonlinear wave equations in an exterior domain in two space dimensions

In this paper we deal with the exterior problem for a system of nonlinear wave equations in two space dimensions, assuming that the initial data is small and smooth. We establish the same type of lower bound of the lifespan for the problem as that for the Cauchy problem, despite of the weak decay property of the solution in two space dimensions.     

Blow-up of solutions for some non-linear wave equations of Kirchhoff type with some dissipation

The initial boundary value problem for non-linear wave equations of Kirchhoff type with dissipation in a bounded domain is considered. We prove the blow-up of solutions for the strong dissipative term -Δ_ut$-\mathrm{\Delta }{u}_t$ and the linear dissipative term _ut$u_t$ by the energy method and give some estimates for the life span of solutions. We also show the nonexistence of global solutions with positive initial energy for non-linear dissipative term by Vitillaro's argument.     

Radially symmetric weak solutions for a quasilinear wave equation in two space dimensions

We prove the convergence of the radially symmetric solutions to the Cauchy problem for the viscoelasticity equations

     

Interaction of kinks for semilinear wave equations with a small parameter

We consider a class of semilinear wave equations with a small parameter and nonlinearities such that the equations have exact kink-type solutions. The main result consists in obtaining sufficient conditions for the nonlinearities under which the interaction of kinks preserves the sine-Gordon scenario. This means that the interaction occurs without changing the waves shape and with shifts of trajectories.     

Global and periodic solutions for nonlinear wave equations with some localized nonlinear dissipation

We discuss the existence of global or periodic solutions to the nonlinear wave equation with the boundary condition , where Ω is a bounded domain in R^N,ρ(x,v) is a function like ρ(x,v)=a(x)g(v) with g′(v)?0 and β(x,u) is a source term of power nonlinearity. a(x) is assumed to be positive only in a neighborhood of a part of the boundary ∂Ω and the stability property is very delicate, which makes the problem interesting.     

Global existence and decay for nonlinear dissipative wave equations with a derivative nonlinearity

We prove the global existence of the so-called H² solutions for a nonlinear wave equation with a nonlinear dissipative term and a derivative type nonlinear perturbation. To show the boundedness of the second order derivatives we need a precise energy decay estimate and for this we employ a ‘loan’ method.     

Global attractors for nonlinear wave equations with nonlinear dissipative terms

We show the existence, size and some absorbing properties of global attractors of the nonlinear wave equations with nonlinear dissipations like ρ(x,ut)=a(x)r|ut|ut.     

Remarks on the energy decay for nonlinear wave equations with nonlinear localized dissipative terms

We derive an energy decay estimate for solutions to the initial-boundary value problem of a semilinear wave equation with a nonlinear localized dissipation. To overcome a difficulty related to derivative-loss mechanism we employ a ‘loan’ method.     

Critical exponent for damped wave equations with nonlinear memory

We consider the Cauchy problem in Rⁿ,n≥1, for a semilinear damped wave equation with nonlinear memory. Global existence and asymptotic behavior as t→∞ of small data solutions have been established in the case when 1≤n≤3. We also derive a blow-up result under some positive data in any dimensional space.     

Global singularity structures of weak solutions to 4-D semilinear dispersive wave equations

In this paper, we are concerned with the global singularity structures of weak solutions to 4-D semilinear dispersive wave equations whose initial data are chosen to be discontinuous on the unit sphere. Combining Strichartz's inequality with the commutator argument techniques, we show that the weak solutions are C²−regular away from the focusing cone surface |x|=|t−1| and the outgoing cone surface |x|=t+1. This research was supported by the National Natural Science Foundation of China and the Doctoral Foundation of NEM of China.     

Global existence and blow up of solutions to systems of nonlinear wave equations with degenerate damping and source terms

We focus on the global well-posedness of the system of nonlinear wave equations

     

On the solvability of one multidimensional version of the first Darboux problem for some nonlinear wave equations

In the present paper, for wave equations with power nonlinearity we investigate the problem of the existence or nonexistence of global solutions of a multidimensional version of the first Darboux problem in the conic 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.     

Regularity of invariant sets in semilinear damped wave equations

Under fairly general assumptions, we prove that every compact invariant subset I of the semiflow generated by the semilinear damped wave equation

     

Remarks on a weighted energy estimate and its application to nonlinear wave equations in one space dimension

A weighted energy estimate with tangential derivatives on the light cone is applied for the Cauchy problem of semilinear wave equations with the null conditions in one space dimension. The well-posedness and lifespan of the solutions are considered based on the vector field method.     

Global energy conservative solutions to a system of variational wave equations

This paper is concerned with a system of variational wave equations which is the Euler–Lagrange equations of a variational principle arising in the theory of nematic liquid crystals and a few other physical contexts. The global existence of an energy-conservative weak solution to its Cauchy problem for initial data of finite energy is established by using the method of energy-dependent coordinates and the Young measure theory.     

Strongly damped wave equations with critical nonlinearities

This note deals with the strongly damped nonlinear wave equation

_utt−Δ_ut−Δu+f(_ut)+g(u)=h$u_{tt}-\Delta u_t-\Delta u+f\left(u_t\right)+g\left(u\right)=h$

with Dirichlet boundary conditions, where both the nonlinearities f$f$ and g$g$ exhibit a critical growth, while h$h$ is a time-independent forcing term. The existence of an exponential attractor of optimal regularity is proven. As a corollary, a regular global attractor of finite fractal dimension is obtained.