2026-06-27T06:06:03Zhttps://docta.ucm.es/rest/oai/request

oai:docta.ucm.es:20.500.14352/577632023-08-11T07:45:35Zcom_20.500.14352_14col_20.500.14352_15

Boundedness and blow up for a semilinear reaction-diffusion system Escobedo, M. Herrero, Miguel A. We consider the semilinear parabolic system (S) { ut-Δu=vp ; vt-Δv=uq, where x Є R(N) (N ≥ 1), t > 0, and p, q are positive real numbers. At t=0, nonnegative, continuous, and bounded initial values (u0(x), 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 S(T)= [0, T) x R(N), 0 < T ≤ ∞. Set T* = sup {T> 0 : u, v remain bounded in S(T)}. 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 + l)(pq - 1) < N/2 both situations coexist, since some nontrivial solutions remain bounded in any strip S(T) while others exhibit finite blow up times. 2023-06-20T17:05:49Z 2023-06-20T17:05:49Z 1991-01 journal article 0022-0396 10.1016/0022-0396(91)90118-S https://hdl.handle.net/20.500.14352/57763 http://www.sciencedirect.com/science/article/pii/002203969190118S http://www.sciencedirect.com eng G.V. 127310-1/87 PB86-0112-CO202 restricted access Elsevier