An existence and uniqueness theorem for a second order nonlinear system

In this paper we give an existence and uniqueness theorem for a nonlinear second order homogeneous Dirichlet system.     

Multiple solutions for a semilinear boundary value problem: a computational multiplicity proof

We prove the first genuine “partial differential equation” result on a conjecture concerning the number of solutions of second-order elliptic boundary value problems with a nonlinearity which grows superlinearly at +∞. The proof makes massive use of computer assistance: After approximate solutions have been computed by a numerical mountain pass algorithm, combined with a Newton iteration to improve accuracy, a fixed point argument is used to show the existence of exact solutions close to the approximations.     

Boundedness, global existence and continuous dependence for nonlinear dynamical systems describing physiologically structured populations

The paper is aimed as a contribution to the general theory of nonlinear infinite dimensional dynamical systems describing interacting physiologically structured populations. We carry out continuation of local solutions to maximal solutions in a functional analytic setting. For maximal solutions we establish global existence via exponential boundedness and by a contraction argument, adapted to derive uniform existence time. Moreover, within the setting of dual Banach spaces, we derive results on continuous dependence with respect to time and initial state.To achieve generality the paper is organized top down, in the way that we first treat abstract nonlinear dynamical systems under very few but rather strong hypotheses and thereafter work our way down towards verifiable assumptions in terms of more basic biological modelling ingredients that guarantee that the high level hypotheses hold.     

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.     

Uniqueness implies existence and uniqueness conditions for nonlocal boundary value problems for nth order differential equations

For the nth order differential equation, y(n)=f(x,y,y^′,…,y(n−1)), we consider uniqueness implies existence results for solutions satisfying certain nonlocal (k+2)-point boundary conditions, 1?k?n−1. Uniqueness of solutions when k=n−1 is intimately related to uniqueness of solutions when 1?k?n−2. These relationships are investigated as well.     

Global existence for the full von Kármán system

We consider here the full system of dynamic von Kármán equations, taking into account the in-plane acceleration terms, which is a model for the vibrations of a nonlinear elastic plate. We prove global existence and uniqueness of strong solutions for this system with various boundary conditions possibly including feedback terms which are useful for stabilization purposes.     

Multiple existence of solutions for a semilinear elliptic problem with Neumann boundary condition

The multiplicity of solutions for semilinear elliptic equations with exponential growth nonlinearities is treated. The approach to the problem is a variational method.     

Global existence of solutions for semilinear damped wave equation in 2-D exterior domain

We consider a mixed problem of a damped wave equation u_tt−Δu+u_t=|u|^p in the two dimensional exterior domain case. Small global in time solutions can be constructed in the case when the power p on the nonlinear term |u|^p satisfies p∗=2<p<+∞. For this purpose we shall deal with a radially symmetric solution in the exterior domain. A new device developed in Ikehata-Matsuyama (Sci. Math. Japon. 55 (2002) 33) plays an effective role.     

非线性退缩抛物组初边值问题解的存在性和唯一性

本文讨论一类非线性退缩抛物型方程组初边值问题广义问题广义解的存在性和唯一性。     

Existence and uniqueness of solutions for third-order nonlinear boundary value problems

In this paper, we present some general results of existence and uniqueness of solutions of nonlinear two-point boundary value problems for third-order nonlinear differential equations by using the Shooting method. As applications we give certain concrete sufficient conditions for the existence and uniqueness.     

Almost global existence for the initial value problem of nonlinear elastodynamic system

The authors deal with the Cauchy problem with small initial data for the nonlinear elastodynamic system. The almost global existence of solution to this problem is proved in a simpler way and a lower bound for the lifespan of solutions is given.     

Existence and uniqueness of positive solutions for a class of semilinear elliptic systems

The authors prove the uniqueness and existence of positive solutions for the semilinear elliptic system which involves nonlinearities with sublinear growth conditions.     

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.     

Global existence of solutions for the heat equation with a nonlinear boundary condition

We consider the initial-boundary value problem for the heat equation with a nonlinear boundary condition:

     

Global existence and nonexistence of solution for Cauchy problem of multidimensional double dispersion equations

In this paper we consider the Cauchy problem of multidimensional generalized double dispersion equations utt−Δu−Δutt+Δ²u=Δf(u), where f(u)=ap|u|. By potential well method we prove the existence and nonexistence of global weak solution without establishing the local existence theory. And we derive some sharp conditions for global existence and lack of global existence solution.     

Global existence and nonexistence of solution for Cauchy problem of two-dimensional generalized Boussinesq equations

In this paper we consider the Cauchy problem of two-dimensional generalized Boussinesq-type equation _utt−Δu−Δ_utt+Δ²u+Δf(u)=0$u_{tt}-\mathrm{\Delta }u-\mathrm{\Delta }{u}_{tt}+{\mathrm{\Delta }}^{2}u+\mathrm{\Delta }f(u)=0$. Under the assumption that f(u)$f(u)$ is a function with exponential growth at infinity and under some assumptions on the initial data, we prove the existence and nonexistence of global weak solution. There are very few works on Boussinesq equation with nonlinear exponential growth term by potential well theory.     

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