Hyperbolic-parabolic singular perturbation for mildly degenerate Kirchhoff equations: Time-decay estimates

We consider the second order Cauchy problem

     

Spectral gap global solutions for degenerate Kirchhoff equations

We consider the second order Cauchy problem

     

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 existence and decay for the semilinear thermoelastic contact problem

In this paper a class of semilinear thermoelastic contact problems is considered and the existence and exponential decay of the weak solutions are obtained.     

Global solutions for dissipative Kirchhoff strings with non-Lipschitz nonlinear term

We investigate the evolution problem

     

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 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.     

A uniqueness result for Kirchhoff equations with non-Lipschitz nonlinear term

We consider the second order Cauchy problem

     

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.     

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.     

Global existence and uniform decay for a nonlinear viscoelastic equation with damping

In this paper we investigate a nonlinear viscoelastic equation with linear damping. Global existence of weak solutions and the uniform decay estimates for the energy have been established.     

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 existence and uniform decay of a damped Klein-Gordon equation in a noncylindrical domain

In this paper, we consider a damped Klein-Gordon equation in a noncylindrical domain. This work is devoted to proving the existence of global solutions and decay for the energy of solutions for a damped Klein-Gordon equation in a noncylindrical domain.     

Global existence and asymptotic convergence of weak solutions for the one-dimensional Navier-Stokes equations with capillarity and nonmonotonic pressure

We construct global weak solution of the Navier-Stokes equations with capillarity and nonmonotonic pressure. The volume variable v₀ is initially assumed to be in H¹ and the velocity variable u₀ to be in L² on a finite interval [0,1]. We show that both variables become smooth in positive time and that asymptotically in time u→0 strongly in L²([0,1]) and v approaches the set of stationary solutions in H¹([0,1]).     

Large time behavior for some nonlinear degenerate parabolic equations

We study the asymptotic behavior of Lipschitz continuous solutions of nonlinear degenerate parabolic equations in the periodic setting. Our results apply to a large class of Hamilton–Jacobi–Bellman equations. Defining Σ as the set where the diffusion vanishes, i.e., where the equation is totally degenerate, we obtain the convergence when the equation is uniformly parabolic outside Σ and, on Σ, the Hamiltonian is either strictly convex or satisfies an assumption similar of the one introduced by Barles–Souganidis (2000) for first-order Hamilton–Jacobi equations. This latter assumption allows to deal with equations with nonconvex Hamiltonians. We can also release the uniform parabolic requirement outside Σ. As a consequence, we prove the convergence of some everywhere degenerate second-order equations.     

On very weak solutions of degenerate p-harmonic equations

We establish a regularity result for very weak solutions of some degenerate elliptic PDEs. The nonnegative function which measures the degree of degeneracy of ellipticity bounds is assumed to be exponentially integrable. We find that the scale of improved regularity is logarithmic.      

Global existence and asymptotic behavior of classical solutions of quasilinear hyperbolic systems with linearly degenerate characteristic fields

In this paper, we study the global existence and the asymptotic behavior of classical solution of the Cauchy problem for quasilinear hyperbolic system with constant multiple and linearly degenerate characteristic fields. We prove that the global C¹ solution exists uniquely if the BV norm of the initial data is sufficiently small. Based on the existence result on the global classical solution, we show that, when the time t tends to the infinity, the solution approaches a combination of C¹ traveling wave solutions. Finally, we give an application to the equation for time-like extremal surfaces in the Minkowski space-time R1+n.