Blowup Estimates for a Semilinear Reaction Diffusion System

Abstract

This paper deals with the blowup estimates of positive solutions for a semilinear reaction diffusion system u t = u + u α v p , v t = v + u q v β , with null Dirichlet boundary conditions. The upper and lower bounds of blowup rates are obtained.

Properties of non-simultaneous blow-up in heat equations coupled via different localized sources

“…We say that the solution (u, v) blows up simultaneously if lim sup t!T kuðÁ; tÞk L 1 ðXÞ ¼ lim sup t!T kvðÁ; tÞk L 1 ðXÞ ¼ þ1. So the non-simultaneous blow-up means that, e.g., lim sup t!T kuðÁ; tÞk L 1 ðXÞ ¼ þ1 with kvðÁ; tÞk L 1 ðXÞ < þ1; t 2 ½0; T. The simultaneous blow-up rate of (1.2) in some exponent region was obtained by Wang [21] and Zheng [23] The other studies about system (1.2) were considered in e.g., [5,7,19,20], where the blow-up criteria, blow-up rate, and even blow-up profile were considered. The non-simultaneous blow-up had been observed and discussed by Quirós and Rossi [15] for the Cauchy problem of (1.2) in R N .…”

Critical exponents and asymptotic estimates of solutions to parabolic systems with localized nonlinear sources

“…2 Remark 3.1. We find from Theorems 3.2 and 3.3 that the blow-up rates of solutions for (1.1) (or (1.2)) with localized reaction terms have the same powers as those for the classical problems (1.5) (see [11,15]) (or (1.6), see [17]). However, other than the classical one (1.5) (or (1.6)), the blow-up set of solutions to the localized system (1.1) (or (1.2)) is the whole domain Ω.…”

“…One can see, e.g., [4,10,16] for the existence and nonexistence of global solutions and [11,15] for the blow-up rates of solutions to (1.5). The blow-up rates of solutions to (1.6) were known already also [17].…”

Blow-up rate estimates for a doubly coupled reaction–diffusion system

“…The blow-up rates of radial solutions to the homogeneous Dirichlet problem of coupled reaction-diffusion equations u t = ∆u + u l 11 v l 12 , v t = ∆v + u l 21 v l 22 in B R × (0, T ) (1.3) were obtained by Zheng [35] and Wang [28] as 22 ) ,β 2 = l 21 + 1 − l 11 l 12 l 21 − (1 − l 11 )(1 − l 22 ) .…”