Displaying similar documents to “Blow up, global existence and growth rate estimates in nonlinear parabolic systems”

Incidence structures of type ( p , n )

František Machala (2003)

Czechoslovak Mathematical Journal

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Every incidence structure 𝒥 (understood as a triple of sets ( G , M , I ) , I ⊆ G × M ) admits for every positive integer p an incidence structure 𝒥 p = ( G p , M p , I p ) where G p ( M p ) consists of all independent p -element subsets in G ( M ) and I p is determined by some bijections. In the paper such incidence structures 𝒥 are investigated the 𝒥 p ’s of which have their incidence graphs of the simple join form. Some concrete illustrations are included with small sets G and M .

Global existence and blow up of solutions for a completely coupled Fujita type system of reaction-diffusion equations

Joanna Rencławowicz (1998)

Applicationes Mathematicae

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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.

The regularisation of the N -well problem by finite elements and by singular perturbation are scaling equivalent in two dimensions

Andrew Lorent (2009)

ESAIM: Control, Optimisation and Calculus of Variations

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Let K : = S O 2 A 1 ∪ S O 2 A 2 ⋯ S O 2 A N where A 1 , A 2 , ⋯ , A N are matrices of non-zero determinant. We establish a sharp relation between the following two minimisation problems in two dimensions. Firstly the N -well problem with surface energy. Let p ∈ 1 , 2 , Ω ⊂ ℝ 2 be a convex polytopal region. Define I ϵ p u = ∫ Ω d p D u z , K + ϵ D 2 u z 2 d L 2 z and let A F denote the subspace of functions in W 2 , 2 Ω that satisfy the affine boundary condition D u = F on ∂ Ω (in the sense of trace), where F ∉ K . We consider the scaling (with respect to ϵ ) of m ϵ p : = inf u ∈ A F I ϵ p u . Secondly the finite...