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Critical Exponents for a System of Heat Equations Coupled in a Non-linear Boundary Condition

✍ Scribed by Bei Hu; Hong-Ming Yin

Publisher
John Wiley and Sons
Year
1996
Tongue
English
Weight
741 KB
Volume
19
Category
Article
ISSN
0170-4214

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✦ Synopsis

In this paper we consider a system of heat equations ut = Au, v, = Av in an unbounded domain R c RN coupled through the Neumann boundary conditions u, = up, v, = up, where p > 0, q > 0, p q > 1 and v is the exterior unit normal on aR. It is shown that for several types of domain there exists a critical exponent such that all of positive solutions blow up in a finite time in subcritical case (including the critical case) while there exist positive global solutions in the supercritical case if initial data are small.

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## Abstract In this paper, we study a system of heat equations $u\_t=\Delta u, \, v\_t=\Delta v\,{\rm in}\,\Omega\times(0,T)$ coupled __via__ nonlinear boundary conditions Here __p__, __q__>0. We prove that the solutions always blow up in finite time for non‐trivial and non‐negative initial value