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The Blow‐up Rate for a System of Heat Equations with Non‐trivial Coupling at the Boundary
Author(s) -
Rossi Julio D.
Publication year - 1997
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/(sici)1099-1476(19970110)20:1<1::aid-mma843>3.0.co;2-e
Subject(s) - mathematics , corollary , ball (mathematics) , mathematical analysis , domain (mathematical analysis) , boundary (topology) , boundary value problem , coupling (piping) , heat equation , mathematical physics , pure mathematics , mechanical engineering , engineering
We study the blow‐up rate of positive radial solutions of a system of two heat equations, ( u 1 ) t =Δ u 1 ( u 2 ) t =Δ u 2 , in the ball B (0, 1), with boundary conditions\documentclass{minimal} \begin{document} \[ \frac{\partial u_1}{\partial r}=(u_1)^{p_{11}}(u_2)^{p_{12}},\qquad \frac{ \partial u_2}{\partial r}=(u_1)^{p_{21}}(u_2)^{p_{22}}. \] \end{document}Under some natural hypothesis on the matrix P =( p ij ) that guarrantee the blow‐up of the solution at time T , and some assumptions of the initial data u 0 i , we find that if ∥ x 0 ∥=1 then u i ( x 0 , t ) goestoinfinitylike( T − t ) α i /2 , where the α i <0 are the solutions of ( P −Id)(α 1 ,α 2 ) t =(−1,−1) t . As a corollary of the blow‐up rate we obtain the loclaization of the blow‐up set at the boundary of the domain. © 1997 by B.G. Teubner Stuttgart‐John Wiley & Sons, Ltd.