Boundedness and blow up for a semilinear reaction-diffusion system

Abstract We consider the semilinear parabolic system (S) u t − Δu = ν p ν t − Δν = u q , 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) × R N, 0 T∗ = sup {T > 0:u, v remain bounded in S T } . We show in this paper that if 0 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.