ISSN:
0170-4214
Keywords:
Mathematics and Statistics
;
Applied Mathematics
Source:
Wiley InterScience Backfile Collection 1832-2000
Topics:
Mathematics
Notes:
In this paper we condiser non-negative solutions of the initial value problem in ∝N for the system \documentclass{article}\pagestyle{empty}\begin{document}$$ {\rm u}_{{\rm t = }} {\rm \delta \Delta u + v}^{\rm p}, $$\end{document} \documentclass{article}\pagestyle{empty}\begin{document}$$ {\rm v}_{{\rm t = }} {\rm \Delta v + u}^{\rm q}, $$\end{document} where 0 ≤ δ ≤ 1 and pq 〉 0. We prove the following conditions.Suppose min(p,q)≥1 but pq1.(a)If δ = 0 then u=v=0 is the only non-negative global solution of the system.(b)If δ〉0, non-negative non-globle solutions always exist for suitable initial values.(c)If 0〈≤1 and max(α, β) ≥ N/2, where qα = β + 1, pβ = α + 1, then the conclusion of (a) holds.(d)If N 〉 2, 0 〈 δ ≤ 1 and max (α β) 〈 (N - 2)/2, then global, non-trivial non-negative solutions exist which belong to L∞(∝N×[0, ∞]) and satisfy 0 〈 u(X, t) ≤ c∣x∣-2α and 0 〈 v(X, t) ≤ c ∣x∣-2bT for large ∣x∣ for all t 〉 0, where c depends only upon the initial data.(e)Suppose 0 〉 δ 1 and max (α, β) 〈 N/2. If N〉 = 1,2 or N 〉 2 and max (p, q)≤ N/(N-2), then global, non-trivial solutions exist which, after makinng the standard ‘hot spot’ change of variables, belong to the weighted Hilbert space H1 (K) where K(x) ≡ exp(¼∣x∣2). They decay like e[max(α,β)-(N/2)+ε]t for every ε 〉 0. These solutions are classical solutions for t 〉 0.(f)If max (α, β) 〈 N/2, then threre are global non-tivial solutions which satisfy, in the hot spot variables \documentclass{article}\pagestyle{empty}\begin{document}$$ \max (u,v)(x,t) \le c(u_0,v_0){\rm e}^{ - \frac{1}{4}|x|^2 } {\rm e}^{[\max (\alpha, \beta) - N/2) + \varepsilon]t}, $$\end{document} where where 0 〈 ε = ε(u0, v0) 〈 (N/2)-;max(α, β).Suppose min(p, q) ≤ 1.(g)If pq ≥ 1, all non-negative solutions are global.Suppose min(p, q) 〈 1.(h)If pg 〉 1 and δ = 0, than all non-trivial non-negative maximal solutions are non-global.(i)If 0 〈 δ ≤ 1, pq 〉 1 and max(α,β)≥ N/2 all non-trivial non-negative maximal solutions are non-global.(j)If 0 〈 δ ≥ 1, pq 〉 1 and max(α,β) 〈 N/2, there are both global and non-negative solutions.We also indicate some extensions of these results to moe general systems and to othere geometries.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1002/mma.1670171005
Permalink
