Criteria For Global Existence Research Articles

Existence of Solutions for the Initial Value Problem with Hadamard Fractional Derivatives in Locally Convex Spaces

In this paper, we investigate an initial value problem for a nonlinear fractional differential equation on an infinite interval. The differential operator is taken in the Hadamard sense and the nonlinear term involves two lower-order fractional derivatives of the unknown function. In order to establish the global existence criteria, we first verify that there exists a unique positive solution to an integral equation based on a class of new integral inequality. Next, we construct a locally convex space, which is metrizable and complete. On this space, applying Schäuder’s fixed point theorem, we obtain the existence of at least one solution to the initial value problem.

Sharp criterion of global existence and orbital stability of standing waves for the nonlinear Schrödinger equation with partial confinement.

In this article, we consider the global existence and stability issues of the nonlinear Schrödinger equation with partial confinement. First, by establishing some new cross-invariant manifolds and variational problems, a new sharp criterion of global existence is derived in the $ L^{2} $-critical and $ L^{2} $-supercritical cases. Then, the existence of orbitally stable standing waves is obtained in the $ L^{2} $-subcritical and $ L^{2} $-critical cases by taking advantage of the profile decomposition technique. Our work extends and complements some earlier results.

Global existence, blow-up in finite time and vacuum isolating phenomena for a system of semilinear wave equations associated with the helical flows of Maxwell fluid

In this paper, we investigate the initial boundary value problem for the system of semilinear wave equations associated with the helical flows of Maxwell fluid. We introduce a family of potential wells and discuss the invariant sets and vacuum isolating behavior of solutions. Using the potential well argument, we show the global existence, finite time blow-up, the asymptotic behavior of solutions, estimate the lifespan as well as blow-up rate of the weak solution. In particular, we establish a sharp criterion for global existence and blow-up of solutions when E0⩽d. Finally, when the initial energy is supercritical, we give some explicit criterion for blow-up in finite.

On blowup solutions for the mixed fractional Schrödinger equation of Choquard type

We investigate some mixed fractional nonlinear Schrödinger equation of Choquard type. In the subcritical energy case local well-posedness is obtained. Moreover, based on a suitable Virial type inequality we show a general criterion of global existence versus blow-up for mass super-critical solutions.

Blowup property of solutions in the parabolic equation with p-Laplacian operator and multi-nonlinearities

ABSTRACT In this paper, we study blowup properties of weak solutions in their norm to the degenerate parabolic equation with multi-nonlinearities and gradient terms. First, we show the existence and uniqueness of weak solutions by using the priori estimate methods. Second, we obtain the global existence criteria and blowup criteria after proving some gradient estimates for different coefficients. Third, we use some Sobolev's inequalities and deal with some differential inequalities of new barrier functions to determine some upper and lower bounds of blowup time of solutions. It could be found out that the blowup or global existence phenomena depend sensitively on the relationship among the different exponents of nonlinearities, which discover a key clue to the different effect of diffusion, gradient term, source term and absorption term on the singular properties of weak solutions.

A Cauchy problem for fractional evolution equations with Hilfer’s fractional derivative on semi-infinite interval

In this paper, we consider a Cauchy problem for fractional evolution equations with Hilfer’s fractional derivative on semi-infinite interval. An elementary fact shows that semi-infinite interval is not compact, the classical Ascoli-Arzelà theorem is not valid. In order to establish the global existence criteria, we first generalize Ascoli-Arzelà theorem into the semi-infinite interval. Next, we introduce a new concept of mild solutions based on cosine/sine family and probability density function and obtain several existence results of mild solutions on semi-infinite interval.

Focusing intercritical NLS with inverse-square potential

We study the focusing nonlinear Schrödinger equation with inverse-square potential with and in three dimensions. In this paper we obtain blow-up and global existence criteria for the solution to () by the method from Duyckaerts-Roudenko [1], which extend a recent work of Killip-Murphy-Visan-Zheng [2] to above the threshold.

Global solvability criteria for quaternionic Riccati equations

Some global existence criteria for quaternionic Riccati equations are established. Two of them are used to prove a completely non conjugation theorem for solutions of linear systems of ordinary differential equations.

Normalized ground states for the NLS equation with combined nonlinearities

We study existence and properties of ground states for the nonlinear Schrödinger equation with combined power nonlinearities−Δu=λu+μ|u|q−2u+|u|p−2uin RN, N≥1, having prescribed mass∫RN|u|2=a2. Under different assumptions on q<p, a>0 and μ∈R we prove several existence and stability/instability results. In particular, we consider cases when2<q≤2+4N≤p<2⁎,q≠p, i.e. the two nonlinearities have different character with respect to the L2-critical exponent. These cases present substantial differences with respect to purely subcritical or supercritical situations, which were already studied in the literature.We also give new criteria for global existence and finite time blow-up in the associated dispersive equation.

Well-Posedness and Stability Results for Some Periodic Muskat Problems

We study the two-dimensional Muskat problem in a horizontally periodic setting and for fluids with arbitrary densities and viscosities. We show that in the presence of surface tension effects the Muskat problem is a quasilinear parabolic problem which is well-posed in the Sobolev space H^r({mathbb {S}}) for each rin (2,3). When neglecting surface tension effects, the Muskat problem is a fully nonlinear evolution equation and of parabolic type in the regime where the Rayleigh–Taylor condition is satisfied. We then establish the well-posedness of the Muskat problem in the open subset of H^2({mathbb {S}}) defined by the Rayleigh–Taylor condition. Besides, we identify all equilibrium solutions and study the stability properties of trivial and of small finger-shaped equilibria. Also other qualitative properties of solutions such as parabolic smoothing, blow-up behavior, and criteria for global existence are outlined.

Blowup results and concentration in focusing Schrödinger-Hartree equation

<p style='text-indent:20px;'>This paper is concerned with the Cauchy problem of the Schrödinger-Hartree equation. Applying the profile decomposition of bounded sequence in <inline-formula><tex-math id="M1">\begin{document}$ \dot{H}^1(\mathbb{R}^N)\cap\dot{H}^{S_c}(\mathbb{R}^N) $\end{document}</tex-math></inline-formula> and corresponding variational structure, a refined Gagliardo-Nirenberg inequality is established and the sharp constant for this inequality is deduced. Secondly, via construction and analysis of some invariant manifolds, we derive a different criterion of global existence and blowup results. Under the discussion of Bootstrap argument, we additionally obtain other sufficient condition for global existence. Finally, A compactness result is applied to show that the blowup solutions with bounded <inline-formula><tex-math id="M2">\begin{document}$ \dot{H}^{S_c} $\end{document}</tex-math></inline-formula> norm definitely have concentration properties related to a fixed <inline-formula><tex-math id="M3">\begin{document}$ \dot{H}^{S_c} $\end{document}</tex-math></inline-formula> norm of certain standing waves.

On the global well-posedness of the inviscid generalized Proudman–Johnson equation using flow map arguments

We reformulate the Generalized Proudman–Johnson (GPJ) equation with parameter a in Lagrangian variables, where it takes the form of an inhomogeneous Liouville equation. This allows us to provide an explicit formula for the flow map (explicit up to the solution of an ODE). Depending on the parameter a, we prove new criteria for global existence or formation of a finite-time singularity and re-derive results from the literature. In particular, we show that there exist smooth initial data which become singular in finite time for a>1. We also give a physical derivation of the GPJ equation for general parameter values of a.

Global existence for a class of viscous systems of conservation laws

We prove existence and boundedness of classical solutions for a family of viscous conservation laws in one space dimension for arbitrarily large time. The result relies on H. Amann’s criterion for global existence of solutions and on suitable uniform-in-time estimates for the solution. We also apply Jüngel’s boundedness-by-entropy principle in order to obtain global existence for systems with possibly degenerate diffusion terms. This work is motivated by the study of a physical model for the space-time evolution of the strain and velocity of an anharmonic spring of finite length.

Quenching Phenomena Due to a Concentrated Nonlinear Source in an Infinitely Long Cylinder

This article studies a semilinear parabolic first initial-boundary value problem with a concentrated nonlinear source in an infinitely long cylinder. We study the effects of the strength of the source on quenching. Criteria for global existence of the solution and for quenching are investigated.

Global existence and quenching for a damped hyperbolic MEMS equation with the fringing field

We study a damped hyperbolic MEMS equation with the fringing field.It arises in the Micro-Electro Mechanical System(MEMS) devices. We give some criteria for global existence and quenching of the solution.First we establish a time-local solution by a contraction mapping theorem. This procedure is standard.Next we show that there exists a global solution for the small parameter and initial value. Finally, we deal with the quenching result for the large parameter.

Time‐weighted blow‐up rates and pointwise profile for single‐point blow‐up solutions in reaction–diffusion equations

This paper deals with asymptotic behavior for blow‐up solutions to time‐weighted reaction–diffusion equations ut=Δu+eαtvp and vt=Δv+eβtuq, subject to homogeneous Dirichlet boundary. The time‐weighted blow‐up rates are defined and obtained by ways of the scaling or auxiliary‐function methods for all α, . Aiding by key inequalities between components of solutions, we give lower pointwise blow‐up profiles for single‐point blow‐up solutions. We also study the solutions of the system with variable exponents instead of constant ones, where blow‐up rates and new blow‐up versus global existence criteria are obtained. Time‐weighted functions influence critical Fujita exponent, critical Fujita coefficient and formulae of blow‐up rates, but they do not limit the order of time‐weighted blow‐up rates and pointwise profile near blow‐up time. Copyright © 2017 John Wiley &amp; Sons, Ltd.

Nonlinear Hartree equation in high energy-mass

This paper is concerned with the Cauchy problem of the nonlinear Hartree equation. By constructing a constrained variational problem, we get a refined Gagliardo–Nirenberg inequality and the best constant for this inequality. We thus derive two conclusions. Firstly, by establishing and analyzing the invariant manifolds, we obtain a new criteria for global existence and blowup of the solutions. Secondly, we get other sufficient condition for global existence with the discussing of the Bootstrap argument. And based on these two conclusions, we also deduce so-called energy-mass control maps, which expose the relationship between the initial data and the solutions.

A differential equation with state-dependent delay from cell population biology

We analyze a differential equation, describing the maturation of a stem cell population, with a state-dependent delay, which is implicitly defined via the solution of an ODE. We elaborate smoothness conditions for the model ingredients, in particular vital rates, that guarantee the existence of a local semiflow and allow to specify the linear variational equation. The proofs are based on theoretical results of Hartung et al. combined with implicit function arguments in infinite dimensions. Moreover we elaborate a criterion for global existence for differential equations with state-dependent delay. To prove the result we adapt a theorem by Hale and Lunel to the C1-topology and use a result on metric spaces from Diekmann et al.

Yet another criterion for global existence in the 3D relativistic Vlasov–Maxwell system

We prove that solutions of the 3D relativistic Vlasov–Maxwell system can be extended, as long as the quantity σ−1(t,x)=max|ω|=1⁡∫R3dp1+p21(1+v⋅ω)f(t,x,p) is bounded in Lx2.

A stochastic predator-prey model with delays

A stochastic delay predator-prey system is considered. Sufficient criteria for global existence, stochastically ultimately bounded in mean and almost surely asymptotic properties are obtained.