Criteria For Global Existence Research Articles

Energy criteria of global existence for the coupled nonlinear Schrödinger equations with harmonic potentials

This paper is concerned with the coupled nonlinear Schrodinger equations with harmonic potentials which describe the Bose-Einstein condensate under the magnetic trap. Two types of the invariant evolution flows are obtained, then a sharp criterion of global existence and blowing up of solutions for the equations is given in terms of the Hamiltonian invariants. Furthermore, we would answer: How small are the initial data such that the global solutions of the system exist?

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.

Energy criterion of global existence for supercritical nonlinear Schrödinger equation with harmonic potential

This paper is concerned with the supercritical nonlinear Schrödinger equation with a harmonic potential which describes the attractive Bose-Einstein condensate under a magnetic trap. We establish two types of new invariant evolution flows. Then, in terms of the Hamiltonian invariants, we derive a new sharp energy criterion for global existence and blowing up of solutions of the equation, which can be precisely computed in the Bose-Einstein condensate.

On global existence for the spherically symmetric Einstein-Vlasov system in Schwarzschild coordinates

The spherically symmetric Einstein-Vlasov system in Schwarzschild coordinates (i.e. polar slicing and areal radial coordinate) is considered. An improved continuation criterion for global existence of classical solutions is given. Two other types of criteria which prevent finite time blow-up are also given.

Sharp criterion of global existence for a class of nonlinear Schrödinger equation with critical exponent

This paper discusses the nonhomogeneous nonlinear Schrödinger equation with critical exponent. By constructing a constrained variational problem and so-called invariant manifolds of the evolution flow, the authors derive a sharp criterion for blowup and global existence of the solutions.

Quasilinear parabolic problems via maximal regularity

We use maximal $L_p$~regularity to study quasilinear parabolic evolution equations. In contrast to all previous work we only assume that the nonlinearities are defined on the space in which the solution is sought for. It is shown that there exists a unique maximal solution depending continuously on all data, and criteria for global existence are given as well. These general results possess numerous applications, some of which will be discussed in separate publications.

Quasilinear parabolic problems via maximal regularity

We use maximal $L_p$~regularity to study quasilinear parabolic evolution equations. In contrast to all previous work we only assume that the nonlinearities are defined on the space in which the solution is sought for. It is shown that there exists a unique maximal solution depending continuously on all data, and criteria for global existence are given as well. These general results possess numerous applications, some of which will be discussed in separate publications.

Existence and nonexistence of global solutions of some non-local degenerate parabolic systems

This paper establishes a new criterion for global existence and nonexistence of positive solutions of the non-local degenerate parabolic system u t a m p ; = v p ( Δ u + a ∫ Ω v d x ) , v t a m p ; = u q ( Δ v + b ∫ Ω u d x ) , x ∈ Ω , t > 0 , \begin{align*} u_t&=v^p\left (\Delta u+a\int _\Omega v dx\right ), v_t&=u^q\left (\Delta v+b\int _\Omega u dx\right ),\quad x\in \Omega , t>0, \end{align*} with homogeneous Dirichlet boundary conditions, where Ω ⊂ R N \Omega \subset \mathbb {R}^N is a bounded domain with a smooth boundary ∂ Ω \partial \Omega and p , q , a , b p, q, a, b are positive constants. For all initial data, it is proved that there exists a global positive solution iff ∫ Ω φ ( x ) d x ≤ 1 / a b \int _\Omega \varphi (x) dx\leq 1/\sqrt {ab} , where φ ( x ) \varphi (x) is the unique positive solution of the linear elliptic problem − Δ φ ( x ) = 1 , x ∈ Ω ; φ ( x ) = 0 , x ∈ ∂ Ω . -\Delta \varphi (x)=1, x\in \Omega ; \varphi (x)=0, x\in \partial \Omega .

Quenching for a diffusive equation with a concentrated singularity

The diffusion equation with a concentrated singular reaction $v_t=v_{xx} + \epsilon\delta(x-a)f(v)$ $(\epsilon>0, 0<a<1)$ is studied. Criteria for global existence and finite time quenching of the solution are established. The growth rate and estimate on quenching time are also given for a certain class of nonlinearities.

Local and global existence criteria for capillary surfaces in wedges

This paper addresses a conjecture of Concus and Finn [Capillary Wedges Revisited, SIAM J. Math. Anal., in press] on conditions for local existence of solutions of the zero-gravity capillarity equation at a boundary protruding corner pointP of prescribed opening 2α. Geometrically, surfaces of constant mean curvatureH are sought as graphs which meet vertical walls over the boundary in prescribed angles, which are locally constant except for a possible jump discontinuity atP. The conjecture is settled more or less completely in the affirmative, depending on whetherH is to be prescribed. The proof proceeds through a global existence theorem for “moon domains”, which seems of independent interest.

Local and global existence criteria for capillary surfaces in wedges

Local and global existence criteria for capillary surfaces in wedges

An application of Hadamard-Lévy’s theorem to a scalar initial value problem

A global inversion theorem due to Hadamard and Lévy is applied to a Volterra integral operator acting in a space of continuous functions to obtain a one-sided criterion for global existence and uniqueness of a scalar initial value problem.

Bounds for solutions of ordinary differential equations

Abstract : Most of the known explicit bounds, as well as criteria for global existence and boundedness, can be obtained from comparison theorems, together with a detailed analysis of the resulting first order differential equations. The same approach also yields information on the approach of a solution to a limit. The bounds given are sometimes difficult or impossible to calculate explicitly, but it is possible to give slightly weaker bounds, which are more easily calculated. The main conclusion to be drawn is that the comparison method provides not only a powerful tool for obtaining bounds for solutions, but also a unified approach to many such problems. (Author)