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Critical blowup and global existence numbers for a weakly coupled system of reaction-diffusion equations

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Abstract

Let DR N be either all of R n or else a cone in R N whose vertex we may take to be at the origin, without loss of generality. Let p i, qj, i = 1, 2, be nonnegative with 0<p 1+q 1p 2+q 2. We consider the long-time behavior of nonnegative solutions of the system

$$u_t = \Delta u + u^{p_1 } v^{q_1 } , v_t = \Delta v + u^{p_2 } v^{q_2 } $$
((S))

in D × [0, ∞) with u 0 = v 0 = 0 on ∂D, (u, v)t(x,0) = (ν 0, ν 0) t(x), u 0, ν 0≧0, u 0, ν 0 ε L (D). We obtain Fujita-type global existence-global nonexistence theorems for (S) analogous to the classical result of Fujita and others for the initial-value problem for u t = Δu + u p, u(x, 0) = u 0(x) ≧ 0. The principal result in the case D = R N and P 2 q 1 > 0 is that when p 1 ≧ 1, the system behaves like the single equation u t u+u p 1 v q 1 with respect to Fujita-type blowup theorems, whereas if p 1 < 1, the behavior of the system is more complicated. Some of the results extend those of Escobedo & Herrero when D = R N and of Levine when D is a cone. These authors considered (S) in the case of p 1 = q 2 = 0. An example of nonuniqueness is also given.

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Communicated by J. Serrin

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Escobedo, M., Levine, H.A. Critical blowup and global existence numbers for a weakly coupled system of reaction-diffusion equations. Arch. Rational Mech. Anal. 129, 47–100 (1995). https://doi.org/10.1007/BF00375126

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