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Zbl 0906.35044
Huang, Qing; Mochizuki, Kiyoshi; Mukai, Kentaro
Life span and asymptotic behavior for a semilinear parabolic system with slowly decaying initial values.
(English)
[J] Hokkaido Math. J. 27, No.2, 393-407 (1998). ISSN 0385-4035

It is considered the initial value problem for the semilinear parabolic system $$u_t=\Delta u+v^p,\quad v_t=\Delta v+u^q, \quad (x,t)\in \bbfR^N\times \bbfR^+.$$ At $t=0$, nonnegative, bounded and continuous initial values $(u_0(x),v_0(x))$ are prescribed. The main results are for the case when $u_0 \sim (\lambda| x| ^{-\alpha})^{1/(q+1)}$, $v_0 \sim (\lambda| x| ^{-\alpha})^{1/(p+1)}$ with $\lambda>0$, $0\leq a<N\min \{p+1,q+1\}$. The authors consider various questions of global existence and nonexistence, large time behavior or life span of the solutions in terms of simple conditions on $\lambda, a, p, q$ and the space dimension $N$.
[Lubomira Softova (Sofia)]
MSC 2000:
*35K45 Systems of parabolic equations, initial value problems
35K57 Reaction-diffusion equations

Keywords: blow-up; global existence; asymptotic behavior; slowly decaying initial value

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