Backward Blow-up Estimates and Initial Trace for a Parabolic System of Reaction-Diffusion

In this article we study the positive solutions of the parabolic semilinear system of competitive type in Ω × (0, T), where Ω is a domain of ℝ , and p,q > 0, pq ≠ 1, Despite of the lack of comparison principles, we prove local upper estimates in the superlinear case pq > 1 of the form u(x, t)...

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Bibliographic Details
Published in:Advanced nonlinear studies 2010-08, Vol.10 (3), p.707-728
Main Authors: Bidaut-Véron, Marie-Françoise, García-Huidobro, Marta, Yarur, Cecilia
Format: Article
Language:English
Subjects:
backward estimates
competitive systems
initial trace
Parabolic semilinear systems of reaction-diffusion
singularities
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Summary:In this article we study the positive solutions of the parabolic semilinear system of competitive type in Ω × (0, T), where Ω is a domain of ℝ , and p,q > 0, pq ≠ 1, Despite of the lack of comparison principles, we prove local upper estimates in the superlinear case pq > 1 of the form u(x, t) ≦ Ct , v(x, t) ≦ Ct in ω × (0, T ), for any domain ω ⊂⊂ Ω, T ∈ (0, T), and C = C(N, p, q, T , ω). For p, q > 1, we prove the existence of an initial trace at time 0, which is a Borel measure on Ω. Finally we prove that the punctual singularities at time 0 are removable when p, q ≧ 1 + 2/N.
ISSN:1536-1365
2169-0375
DOI:10.1515/ans-2010-0310