BOUNDEDNESS AND BLOW UP FOR A SEMILINEAR REACTION DIFFUSION SYSTEM

We consider the semilinear parabolic system (S) ut - Δu = νp νt - Δν = uq, where x∈RNN ≥ 1, t > 0, and p, q are positive real numbers. At t = 0, nonnegative, continuous, and bounded initial values (u0, 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 ST = [0,T) × RN, 0 < T ≤ ∞. Set T* = sup {T > 0:u, v remain bounded in ST}. 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 + 1)(pq - 1) < N 2 both situations coexist, since some nontrivial solutions remain bounded in any strip SΓ while others exhibit finite blow up times. © 1991.