ALMOST GLOBAL EXISTENCE OF DIRICHLETINITIAL-BOUNDARY VALUE PROBLEM FORNONLINEAR ELASTODYNAMIC SYSTEMOUTSIDE A STAR-SHAPED DOMAIN

This paper deals with the mixed initial-boundary value problem of Dirichlet type for the nonlinear elastodynamic system outside a star-shaped domain. The almost global existence of solution with small initial data to this problem is proved and a lower bound for the lifespan of solutions is given.     

Local existence of solutions for the Neumann problem of the nonlinear elastodynamic system outside a domain

In this paper, we study the mixed initial-boundary value problem of Neumann type for the nonlinear elastic wave equation outside a domain. The local existence of solutions to this problem is proved by iteration. To get this result, we prove the existence of solutions for the second order linear hyperbolic system with variable coefficients (in Sobolev spaces) outside of a domain by using linear evolution operators and integro-differential equations.     

Global existence of solution of Cauchy problem for nonlinear pseudo-parabolic equation

In this paper, we prove that the Cauchy problem for the nonlinear pseudo-parabolic equation

vt−αvxxt−βvxx+γvx+fx(v)=φx(vx)+g(v)−αg(v)_xx     

Global existence of classical solutions to the Cauchy problem for a kind of quasilinear hyperbolic systems in divergence form

This paper deals with the global existence of classical solutions to a kind of second order quasilinear hyperbolic systems subject to a null condition, with the linear elastodynamic system as its principal part and the nonlinear terms depending on the product of u² and the derivatives of u.     

Some remarks on global existence to the Cauchy problem of the wave equation with nonlinear dissipation

In this paper we prove the existence of global decaying H² solutions to the Cauchy problem for a wave equation with a nonlinear dissipative term by constructing a stable set in H¹(?ⁿ ). (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global existence for non‐autonomous Navier–Stokes equations with discontinuous initial data

In this paper, we consider the non‐autonomous Navier–Stokes equations with discontinuous initial data. We prove the global existence of solutions, the decay rate of density, and the equilibrium state of solutions. Copyright © 2017 John Wiley & Sons, Ltd.     

Almost conservation law and global rough solutions to a nonlinear Davey-Stewartson equation

We prove an “almost conservation law” to obtain global-in-time well-posedness for the nonlinear Davey-Stewartson equation in Hs(R²), and .     

Global existence of classical solutions to the minimal surface equation with slow decay initial value

In this paper, we prove that the global existence of classical solutions to the Cauchy problem for the minimal surface equation with slow decay initial value in Minkowskian space time.     

Global existence of weak discontinuous solutions to the Cauchy problem with small BV initial data for quasilinear hyperbolic systems

In this paper, we study the Cauchy problem for quasilinear hyperbolic system with a kind of non‐smooth initial data. Under the assumption that the initial data possess a suitably small bounded variation norm, a necessary and sufficient condition is obtained to guarantee the existence and uniqueness of global weak discontinuous solution. Copyright © 2014 John Wiley & Sons, Ltd.     

Sharp conditions of global existence for the generalized Davey-Stewartson system

This paper discusses the generalized Davey–Stewartsonsystem. By constructing a cross-constrained variational problemand so-called invariant manifolds of the evolution flow, wederive a sharp criterion for blow-up and global existence ofthe solutions.     

Algebraic criteria for global existence or blow-up for a boundary coupled system of nonlinear diffusion equations

This article deals with the critical curves for a degenerate parabolic system coupled via nonlinear boundary flux. By constructing the self-similar supersolution and subsolution, we obtain the critical global existence curve. The critical curve of Fujita type is conjectured with the aid of some new results.     

Remarks on the local existence of solutions to the Debye system

In this paper we study the Cauchy problem of the Debye system for initial data in the Lorentz space Ln,1(Rn) and the Besov space for 0<α<1. Some local existence theorems are proved.     

Global existence, uniqueness, and continuous dependence for a semilinear initial value problem

In this paper, we consider the semilinear initial value problem associated with an operator A whose spectrum lies in a sector of the complex plane and whose resolvent satisfies (z−A)⁻¹M|z|^γ for some −1<γ<0 and all z outside the sector. The properties of existence and uniqueness of global mild solutions and continuous dependence on the initial data are investigated.     

Existence and nonexistence of global solutions to the Cauchy problem for a nonlinear beam equation

The paper studies the existence and nonexistence of global solutions to the Cauchy problem for a nonlinear beam equation arising in the model in variational form for the neo–Hookean elastomer rod where k₁, k₂>0 are real numbers, g(s) is a given nonlinear function. When g(s)=sⁿ (where n?2 is an integer), by using the Fourier transform method we prove that for any T>0, the Cauchy problem admits a unique global smooth solution u∈C^∞((0, T]; H^∞( R ))∩C([0, T]; H³( R ))∩C¹([0, T]; H^?1( R )) as long as initial data u₀∈W^{4, 1}( R )∩H³( R ), u₁∈L¹( R )∩H^?1( R ). Moreover, when (u₀, u₁)∈H²( R ) × L²( R ), g∈C²( R ) satisfy certain conditions, the Cauchy problem has no global solution in space C([0, T]; H²( R ))∩C¹([0, T]; L²( R ))∩H¹(0, T; H²( R )). Copyright © 2009 John Wiley & Sons, Ltd.     

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.     

Sharp criterion of global existence for nonlinear Schrödinger equation with a harmonic potential

This paper discusses nonlinear SchrSdinger equation with a harmonic potential. By constructing a different cross-constrained variational problem and the so-called invariant sets, we derive a new threshold for blow-up and global existence of solutions.     

A highly unstable initial value boundary value problem

In this paper we consider a two-dimensional diffusion equation on the closed right halfspace satisfying a boundary condition for which the minimum principle fails. As a consequence, the associated Cauchy initial value problem fails to be well-posed. In particular, solutions need not exist and, when they do exist, they may do so for only a finite length of time. Among other things, we provide a necessary and sufficient condition on the initial data in order that the solution exist for all time and remain non-negative.