A uniqueness result for a semilinear reaction-diffusion system

Abstract

Let (u(t, x), v(t, x)) and (uBAR(t, x), vBAR(t, x)) be two nonnegative classical solutions of (S)[GRAPHICS:{ut=Δu+vp, p>0 ; vt=Δv+uq, q>0] in some strip S(T) = (0, T) x R(N), where 0 < T ≤ ∞, and suppose that u(0, x) = uBAR(0, x), v(0, x) = vBAR(0, x), where u(0, x) and v(0, x) are continuous, nonnegative, and bounded real functions, one of which is not identically zero. Then one has u(t, x) = uBAR(t, x), v(t, x) = vBAR(t, x) in S(T). If pq ≥ 1, the result is also true if u(0, x) = v(0, x) = 0. On the other hand, when 0 < pq < 1, the set of solutions of (S) with zero initial values is given by u(t; s) = c1(t - s)+(p+1)/(1-pq), v(t; s) = c2(t - s)+(q+1)/(1-qp), where 0 ≤ s ≤ t, c1 and c2 are two positive constants depending only on p and q, and (ξ)+ = max{ξ,0}.