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We examine the parabolic system of three equations $u_{t}$ - Δu = $v^{p}$ , $v_{t}$ - Δv = $w^{q}$ , $w_{t}$ - Δw = $u^{r}$ , x ∈ $ℝ^{N}$ , t > 0 with p, q, r positive numbers, N ≥ 1, and nonnegative, bounded continuous initial values. We obtain global existence and blow up unconditionally (that is, for any initial data). We prove that if pqr ≤ 1 then any solution is global; when pqr > 1 and max(α,β,γ) ≥ N/2 (α, β, γ are defined in terms of p, q, r) then every nontrivial solution exhibits a finite blow up time.

Global existence and blow-up analysis for some degenerate and quasilinear parabolic systems.

Lu, Haihua (2009)

Electronic Journal of Qualitative Theory of Differential Equations [electronic only]

Global existence and blow-up for a completely coupled Fujita type system

Joanna Rencławowicz (2000)

Applicationes Mathematicae

The Fujita type global existence and blow-up theorems are proved for a reaction-diffusion system of m equations (m>1) in the form $u_{i t} = Δ u_{i} + u_{i + 1}^{p_{i}}, i = 1, . . ., m - 1,$ $u_{m t ...}$

Global existence and blow-up for a shallow water equation

Adrian Constantin, Joachim Escher (1998) Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Global existence and convergence of solutions to a cross-diffusion cubic predator-prey system with stage structure for the prey.

Cao, Huaihuo, Fu, Shengmao (2010) Boundary Value Problems [electronic only]

Global existence and convergence to steady states in a chemorepulsion system

Tomasz Cieślak, Philippe Laurençot, Cristian Morales-Rodrigo (2008) Banach Center Publications

In this paper we consider a model of chemorepulsion. We prove global existence and uniqueness of smooth classical solutions in space dimension n = 2. For n = 3,4 we prove the global existence of weak solutions. The convergence to steady states is shown in all cases.

Global existence and decay of solutions of a coupled system of BBM-Burgers equations.

Jardel Morais Pereira (2000) Revista Matemática Complutense

The global well-posedness of the initial-value problem associated to the coupled system of BBM-Burgers equations (*) in the classical Sobolev spaces Hs(R) x Hs(R) for s ≥ 2 is studied. Furthermore we find decay estimates of the solutions of (*) in the norm Lq(R) x Lq(R), 2 ≤ q ≤ ∞ for general initial data. Model (*) is motivated by a work due to Gear and Grimshaw [10] who considered strong interaction of weakly nonlinear long waves governed by a coupled system of KdV equations.

Global existence and energy decay of solutions to a Bresse system with delay terms

Abbes Benaissa, Mostefa Miloudi, Mokhtar Mokhtari (2015) Commentationes Mathematicae Universitatis Carolinae

We consider the Bresse system in bounded domain with delay terms in the internal feedbacks and prove the global existence of its solutions in Sobolev spaces by means of semigroup theory under a condition between the weight of the delay terms in the feedbacks and the weight of the terms without delay. Furthermore, we study the asymptotic behavior of solutions using multiplier method.