The Blow-Up Rate for a Strongly Coupled System of Semilinear Heat Equations with Nonlinear Boundary Conditions

AbstractThe paper deals with the blow-up rate of positive solutions to the system ut=uxx+ul11vl12, vt=vxx+ul21vl22 with boundary conditions ux(1,t)=(up11vp12)(1,t) and 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) (resp. maxx∈[0,1]v(x,t)) goes to infinity as (T−t)α1/2 (resp. (T−t)α2/2), where αi