The blow-up rate for a coupled system of semilinear heat equations with nonlinear boundary conditions

The paper deals with the blow-up rate of positive solutions of the system ut = uxx + ul11vl12, vt = vxx + ul21vl22 with nonlinear boundary conditions ux(0,t) = 0, ux(1,t) = (up11vp12)(1,t), and vx(0,t) = 0, vx(1,t) = (up21vp22)(1,t). Under some assumptions on the matrices L = (lij) and P = (pij) and on the initial data u0, v0, the solution (u,v) blows up at finite time T, and we prove that maxx∈[0,1] u(x,t) (respectively, maxx∈[0,1] v(x,t)) goes to infinity like (T − t)β12 (respectively, (T − t)β22 as t → T, where βi < 0 are the solutions of (L − Id)(β1, β2)t = (−1, −1)t.