Backward Blow-up Estimates and Initial Trace for a Parabolic System of Reaction-Diffusion
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Date
2010
Journal Title
Journal ISSN
Volume Title
Publisher
ADVANCED NONLINEAR STUDIES, INC
Abstract
In this article we study the positive solutions of the parabolic semilinear system of competitive type
{u(t) - Delta u + v(p) = 0, u(t) - Delta v + u(q) = 0,
in Omega x (0, T), where Omega is a domain of R(N), and p, q > 0, pq not equal 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(-(p+1)/(pq-1)), v(x, t) <= Ct(-(q+1)/(pq-1))
in omega x (0, T(1)), for any domain omega subset of subset of Omega, T(1) is an element of (0, T), and C = C(N,p,q,T1,omega). For p, q > 1, we prove the existence of an initial trace at time 0, which is a Borel measure on Omega. Finally we prove that the punctual singularities at time 0 are removable when p,q >= 1+2/N.
{u(t) - Delta u + v(p) = 0, u(t) - Delta v + u(q) = 0,
in Omega x (0, T), where Omega is a domain of R(N), and p, q > 0, pq not equal 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(-(p+1)/(pq-1)), v(x, t) <= Ct(-(q+1)/(pq-1))
in omega x (0, T(1)), for any domain omega subset of subset of Omega, T(1) is an element of (0, T), and C = C(N,p,q,T1,omega). For p, q > 1, we prove the existence of an initial trace at time 0, which is a Borel measure on Omega. Finally we prove that the punctual singularities at time 0 are removable when p,q >= 1+2/N.
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Keywords
Parabolic semilinear systems of reaction-diffusion, competitive systems, backward estimates, initial trace, singularities, POSITIVE SOLUTIONS, ELLIPTIC-SYSTEMS, EQUATIONS, ABSORPTION