Sufficient Conditions For Global Existence Research Articles

Global Hopf bifurcation of a delayed diffusive predator-prey model with Michaelis-Menten-type prey harvesting

This paper is concerned with a delayed diffusive predator–prey model with Michaelis–Menten-type prey harvesting. The model has rich dynamical behaviors. It undergoes transcritical, saddle-node, transcritical saddle-node, Hopf bifurcations, which can cause the behavior mutations of the system, such as the change of the stability, the change of the number of equilibria, and the emergence of the periodic orbits. We first study the stability of the equilibria, and analyze the critical conditions for the above bifurcations at each equilibrium. In addition, the stability and direction of local Hopf bifurcations near the positive steady state are given by the normal form theory and center manifold theorem. Meanwhile, the approximate expressions of periodic orbits are given. Furthermore, we derive the sufficient condition for global existence of the spatial homogeneous periodic orbits. Finally, some numerical simulations are carried out to support our analytical results.

Existence of oscillatory solutions of second order delay differential equations

In this paper, we investigate existence of oscillatory solutions for a forced second order nonlinear delay differential equations [r(t)Φ(x′(t))]′+∑i=1mfi(t,x(gi(t)))=q(t), where fi∈C([t0,∞)×R,R),gi(t)≤t,limt→∞gi(t)=∞,i=1,2,⋯,m,Φ∈C1(R,R),Φ(u) is an increasing function for all u∈R, Φ−1(u) satisfies the local Lipischitz condition. A new sufficient condition for global existence of oscillatory solution is obtained by the Schauder–Tychonoff theorem. When Φ(u)=uα with α≥1 being the ratio of two positive odd integers has also been studied. We give examples to illustrate the applicability of our results.

On global existence for the Gierer-Meinhardt system

We consider the Gierer-Meinhardt system (1.1), shown below, on a bounded smooth domain $\Omega\subset\mathbb{R}^n$ ($n\ge1$) with a homogeneous Neumann boundary condition. For suitable exponents $a$, $b$, $c$ and $d$, we establish certain sufficient conditions for global existence. Theorem 1.1 here, combined with Theorem 1.2 of [6], implies a classical phenomenon on the effect of the initial data on global existence and finite time blow-up. This work is a continuation of our earlier result [6] for the Gierer-Meinhardt system. &nbsp The Gierer-Meinhardt system was introduced in [1] to model activator-inhibitor systems in pattern formation in ecological systems.

Finite time blow-up and global solutions for fourth order damped wave equations

This work is devoted to a class of fourth order wave equations with linear damping term and superlinear source term. After showing the uniqueness and existence of local solutions to the equations, we give necessary and sufficient conditions for global existence and finite time blow-up of these solutions. Moreover, the potential well depth is estimated.

Necessary conditions and sufficient conditions for global existence for two-components nonlinear Schrödinger equations in the super critical case

In this paper, we consider two-components nonlinear Schrödinger equations in the super critical case. We establish a necessary condition and a sufficient condition of global existence of the solution for two-components nonlinear Schrödinger equations. These conditions are charge criterion of global existence in the super critical case, thereby extending the results in the critical case. Furthermore, we improve a blow-up condition.

Global existence and blowup for quasilinear parabolic equations not in divergence form

This paper deals with global and blowup solutions of the quasilinear parabolic equation ut=α(x,u,∇u)Δu+f(x,u,∇u) with homogeneous Dirichlet boundary conditions. We will give sufficient conditions such that the solutions either exist globally or blow up in a finite time for any smooth initial values. In special cases, a necessary and sufficient condition for global existence is given.

ASYMPTOTIC POINTWISE BEHAVIOR FOR SYSTEMS OF SEMILINEAR WAVE EQUATIONS IN THREE SPACE DIMENSIONS

In connection with the weak null condition, Alinhac introduced a sufficient condition for global existence of small amplitude solutions to systems of semilinear wave equations in three space dimensions. We introduce a slightly weaker sufficient condition for the small data global existence, and we investigate the asymptotic pointwise behavior of global solutions for systems satisfying this condition. As an application, the asymptotic behavior of global solutions under the Alinhac condition is also derived.

On the Lagrangian averaged Euler equations: local well-posedness and blow-up criterion

In this article we study local and global well-posedness of the Lagrangian Averaged Euler equations. We show local well-posedness in Triebel-Lizorkin spaces and further prove a Beale-Kato-Majda type necessary and sufficient condition for global existence involving the stream function. We also establish new sufficient conditions for global existence in terms of mixed Lebesgue norms of the generalized Clebsch variables.

GLOBAL EXISTENCE AND BLOW-UP FOR A DOUBLY DEGENERATE PARABOLIC EQUATION SYSTEM WITH NONLINEAR BOUNDARY CONDITIONS

AbstractIn this paper, we deal with the global existence and blow-up of solutions to a doubly degenerative parabolic system with nonlinear boundary conditions. By constructing various kinds of sub- and super-solutions and using the basic properties of M-matrix, we give the necessary and sufficient conditions for global existence of non-negative solutions, which extend the recent results of Zheng, Song and Jiang (S. N. Zheng, X. F. Song and Z. X. Jiang, Critical Fujita exponents for degenerate parabolic equations coupled via nonlinear boundary flux, J. Math. Anal. Appl. 298 (2004), 308–324), Xiang, Chen and Mu (Z. Y. Xiang, Q. Chen, C. L. Mu, Critical curves for degenerate parabolic equations coupled via nonlinear boundary flux, Appl. Math. Comput. 189 (2007), 549–559) and Zhou and Mu (J. Zhou and C. L Mu, On critical Fujita exponents for degenerate parabolic system coupled via nonlinear boundary flux, Pro. Edinb. Math. Soc. 51 (2008), 785–805) to more general equations.

Some nonlinear dynamic inequalities on time scales

Our aim in this paper is to establish some explicit bounds of solutions of a certain class of nonlinear dynamic inequalities on a time scale T which is unbounded above. Also, we establish some sufficient conditions for global existence and an estimate of the rate of decay of the solutions. These on the one hand generalizes and on the other hand furnish a handy tool for the study of qualitative as well as quantitative properties of solutions of dynamic equations on time scales. Mathematics subject classification (2010): 26D15, 26D20, 39A12.

Global existence and nonexistence of the non-Newtonian polytropic filtration equations coupled with nonlinear boundary conditions

In this article, we deal with the global existence and nonexistence of solutions to the non-Newtonian polytropic filtration equations coupled with nonlinear boundary conditions. By constructing various kinds of sub- and super-solutions and using the basic properties of M-matrix, we give the necessary and sufficient conditions for global existence of nonnegative solutions. The critical curve of Fujita type is conjectured with the aid of some new results, which extend the recent results of Zheng, Song, and Jiang [Critical Fujita exponents for degenerate parabolic equations coupled via nonlinear boundary flux, J. Math. Anal. Appl. 298 (2004), pp. 308–324], Zhou and Mu [Critical curve for a non-Newtonian polytropic filtration system coupled via nonlinear boundary flux, Nonlinear Anal. 68 (2008), pp. 1–11], and Zhou and Mu [Algebraic criteria for global existence or blow-up for a boundary coupled system of nonlinear diffusion equations, Appl. Anal. 86 (2007), pp. 1185–1197] to more general equations.

On the motion of polygonal curves with asymptotic lines by crystalline curvature flow with bulk effect

The behavior of polygonal curves with asymptotic lines to crystalline motion with the bulk effect is discussed. We show sufficient conditions for global existence of the solutions and characterize facet-extinction patterns. We also show the eventual monotonicity of shape of the solution curves, that is, the solutions become V-shaped in finite time.

A nonlinear diffusion system coupled via nonlinear boundary flux

In this paper, we deal with the global existence and nonexistence of solutions to a nonlinear diffusion system coupled via nonlinear boundary flux. By constructing various kinds of sub- and super-solutions and using the basic properties of M-matrix, we give the necessary and sufficient conditions for global existence of nonnegative solutions. The critical curve of Fujita type is conjectured with the aid of some new results, which extend the recent results of Wang et al. [Nonlinear Anal. 71 (2009) 2134–2140] and Li et al. [J. Math. Anal. Appl. 340 (2008) 876–883] to more general equations.

On Blow-up Solutions to the 3D Cubic Nonlinear Schrodinger Equation

For the 3d cubic nonlinear Schrodinger (NLS) equation, which has critical (scaling) norms L 3 and u H 1/2 , we first prove a result establishing sufficient conditions for global existence and sufficient conditions for finite-time blow-up. For the rest of the paper, we focus on the study of finite-time radial blow-up and prove a result on the concentration of the L 3 norm at the origin. Two disparate possibilities emerge, one which coincides with solutions typically observed in numer- ical experiments that consist of a specific bump profile with maximum at the origin and focus toward the origin at rate ∼ (T − t) 1/2 , where T > 0 is the blow-up time. For the other possibility, we propose the existence of contracting sphere blow-up solutions, i.e. those that concentrate on a sphere of radius ∼ (T −t) 1/3 , but focus towards this sphere at a faster rate ∼ (T − t) 2/3 . These conjectured solutions are analyzed through heuristic arguments and shown (at this level of precision) to be consistent with all conservation laws of the equation.

Some nonlinear dynamic inequalities on time scales and applications

Some nonlinear dynamic inequalities on a time scale T are formulated in this paper. Some sufficient conditions for global existence and an estimate of the rate of decay of solutions are obtained. The results not only unify the results of differential and difference inequalities but can be applied on different types of time scales. These inequalities are of interest in a study of continuous and discrete dynamical systems and nonlinear evolution equations as well as in oscillation theory of dynamic equations on time scales and can be applied to the study of global existence of nonlinear PDE. Some applications illustrating the main results are given.

Global existence and nonexistence for diffusive polytropic filtration equations with nonlinear boundary conditions

In this paper, we deal with the global existence and nonexistence of solutions to a diffusive polytropic filtration system with nonlinear boundary conditions. By constructing various kinds of sub- and super-solutions and using the basic properties of M-matrix, we give the necessary and sufficient conditions for global existence of nonnegative solutions, which extend the recent results of Li et al. (Z Angew Math Phys 60:284–298, 2009) and Wang et al. (Nonlinear Anal 71:2134–2140, 2009) to more general equations and simplify their proofs slightly.

On rough differential equations

We prove that the Itô map, that is the map that gives the solution of a differential equation controlled by a rough path of finite $p$-variation with $p\in [2,3)$ is locally Lipschitz continuous in all its arguments and we give some sufficient conditions for global existence for non-bounded vector fields.

Global existence and asymptotic behavior of solutions to systems of semilinear wave equations in two space dimensions

We consider the Cauchy problem for systems of semilinear wave equations in 2D with small initial data, and introduce a sufficient condition for global existence of small solutions. Our condition is weaker than the null condition for 2D wave equations, and it is motivated by Alinhac's condition for 3D. We also show that some global solutions under our condition are not asymptotically free.

Non-simultaneous versus simultaneous quenching in a coupled nonlinear parabolic system

This paper deals with finite-time quenching for the nonlinear parabolic system with coupled singular absorptions: u t = Δ u − v − p , v t = Δ v − u − q in Ω × ( 0 , T ) subject to positive Dirichlet boundary conditions, where p , q > 0 , Ω is a bounded domain in R N with smooth boundary. We obtain the sufficient conditions for global existence and finite-time quenching of solutions, and then determine the blow-up of time-derivatives and the quenching set for the quenching solutions. As the main results of the paper, a very clear picture is obtained for radial solutions with Ω = B R : the quenching is simultaneous if p , q ≥ 1 , and non-simultaneous if p < 1 ≤ q or q < 1 ≤ p ; if p , q < 1 with R > 2 N , then both simultaneous and non-simultaneous quenching may happen, depending on the initial data. In determining the non-simultaneous quenching criteria of the paper, some new ideas have been introduced to deal with the coupled singular inner absorptions and inhomogeneous Dirichlet boundary value conditions, in addition to techniques frequently used in the literature.

Stability, bifurcation and global existence of a Hopf-bifurcating periodic solution for a class of three-neuron delayed network models

In this paper a system of three delay differential equations representing a Hopfield type general model for three neurons with two-way (bidirectional) time delayed connections between the neurons and time delayed self-connection from each neuron to itself is studied. Delay independent and delay dependent sufficient conditions for linear stability, instability and the occurrence of a Hopf bifurcation about the trivial equilibrium are addressed. The partition of the resulting parametric space into regions of stability, instability, and Hopf bifurcation in the absence of self-connection is realized. To extend the local Hopf branches for large delay values a particular bidirectional delayed tri-neuron model without self-connection is investigated. Sufficient conditions for global existence of multiple non-constant periodic solutions are obtained for such a model using the global Hopf-bifurcation theorem for functional differential equations due to J. Wu and the Bendixson criterion for higher dimensional ordinary differential equations due to Li and Muldowney, and following the approach developed by Wei and Li.