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.     

Energy decay rates for solutions of the wave equations with nonlinear damping in exterior domain

In this paper we study the behaviors of the energy of solutions of the wave equations with localized nonlinear damping in exterior domains.     

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

     

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.     

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.     

Global existence of weak solutions for two-dimensional semilinear wave equations with strong damping in an exterior domain

We consider two-dimensional mixed problems in an exterior domain for a semilinear strongly damped wave equation with a power-type nonlinearity |u|^p${\left|u\right|}^p$. If the initial data have a small weighted energy, we shall derive a global existence and energy decay results in the case when the power p$p$ of the nonlinear term satisfies p>6$p>6$.     

Global solvability for the Kirchhoff equations in exterior domains of dimension three

We consider the initial (boundary) value problem for the Kirchhoff equations in exterior domains or in the whole space of dimension three, and show that these problems admit time-global solutions, provided the norms of the initial data in the usual Sobolev spaces of appropriate order are sufficiently small. We obtain uniform estimates of the L¹(R) norms with respect to time variable at each point in the domain, of solutions of initial (boundary) value problem for the linear wave equations. We then show that the estimates above yield the unique global solvability for the Kirchhoff equations.     

Finite time blow up for critical wave equations in high dimensions

We prove that solutions to the critical wave equation (1.1) with dimension n?4 can not be global if the initial values are positive somewhere and nonnegative. This completes the solution to the famous Strauss conjecture about semilinear wave equations of the form . The rest of the cases, the lower-dimensional case n?3, and the sub or super critical cases were settled many years earlier by the work of several authors.     

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 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.     

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.     

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.     

Local existence of solutions for the initial-boundary value problem of the nonlinear elastodynamic system outside a domain

In this paper we deal with the mixed initial-boundary value problem of Dirichlet type for the nonlinear elastodynamic system outside a domain. The local existence of solutions to this problem is proved by iteration.     

A global subsonic solution to an interacting transonic shock for the self-similar nonlinear wave equation

We present the existence of the subsonic solution to a two-dimensional Riemann problem governed by a self-similar nonlinear wave equation where the boundary of the subsonic region consists of a transonic shock and the sonic circle. Thus the governing equation becomes a free boundary problem on the transonic shock and degenerates on the sonic circle. By utilizing the barrier methods and iterative methods, we show the well-posedness of the transonic shock in the entire subsonic region and thus establish the global solution. This result does not rely on any smallness of Riemann data.     

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.     

Long-time existence for semi-linear Klein-Gordon equations on tori

We use normal forms for Sobolev energy to prove that small smooth solutions of semi-linear Klein-Gordon equations on the torus exist over a larger interval than the one given by local existence theory, for almost every value mass. The gain on the length of the lifespan does not depend on the dimension. The result relies on the fact that the difference of square of two successive distinct eigenvalues of on Td can be bounded from below by a constant.     

Optimal decay rates for solutions of a nonlinear wave equation with localized nonlinear dissipation of unrestricted growth and critical exponent source terms under mixed boundary conditions

This article investigates optimal decay rates for solutions to a semilinear hyperbolic equation with localized interior damping and a source term. Both dissipation and the source are fully nonlinear   and the growth rate of the source map may include critical exponents (for Sobolev’s embedding H¹→L²$H^1\to L^2$). Besides continuity and monotonicity, no growth or regularity assumptions are imposed on the damping. We analyze the system in the presence of Neumann-type boundary conditions including the mixed cases: Dirichlet–Neumann–Robin.