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

Abstract

Let D ⊂ R ^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, q_j, i = 1, 2, be nonnegative with 0<p ₁+q ₁≦p 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 =Δu+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.