Boundedness and blow up for a semilinear reaction-diffusion system



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Escobedo, M. and Herrero, Miguel A. (1991) Boundedness and blow up for a semilinear reaction-diffusion system. Journal of Differential Equations, 89 (1). pp. 176-202. ISSN 0022-0396

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We consider the semilinear parabolic system
(S) { ut-Δu=vp ; vt-Δv=uq,
where x Є R(N) (N ≥ 1), t > 0, and p, q are positive real numbers. At t=0, nonnegative, continuous, and bounded initial values (u0(x), v0(x)) are prescribed. The corresponding Cauchy problem then has a nonnegative classical and bounded solution (u(t, x), v(t, x)) in some strip S(T)= [0, T) x R(N), 0 < T ≤ ∞. Set T* = sup {T> 0 : u, v remain bounded in S(T)}. We show in this paper that if
0 < pq ≤ 1, then T* = + ∞, so that solutions can be continued for all positive times. When pq > 1 and (γ + 1 ) / (pq - 1) ≥ N/2 with γ = max {p, q}, one has T* < + ∞ for every nontrivial solution (u, v). T* is then called the blow up time of the solution under consideration. Finally, if (γ + l)(pq - 1) < N/2 both situations coexist, since some nontrivial solutions remain bounded in any strip S(T) while others exhibit finite blow up times.

Item Type:Article
Uncontrolled Keywords:Heat-equations; parabolic equations; nonexistence; existence
Subjects:Sciences > Mathematics > Differential equations
ID Code:17443
Deposited On:14 Dec 2012 09:22
Last Modified:18 Feb 2019 13:02

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