Blow-up with logarithmic nonlinearities

We study the asymptotic behaviour of nonnegative solutions of the nonlinear diffusion equation in the half-line with a nonlinear boundary condition,{ut=uxx−λ(u+1)logp(u+1),(x,t)∈R+×(0,T),−ux(0,t)=(u+1)logq(u+1)(0,t),t∈(0,T),u(x,0)=u0(x),x∈R+, with p,q,λ>0p,q,λ>0. We describe in terms of p, q and λ when the solution is global in time and when it blows up in finite time. For blow-up solutions we find the blow-up rate and the blow-up set and we describe the asymptotic behaviour close to the blow-up time, showing that a phenomenon of asymptotic simplification takes place. We finally study the appearance of extinction in finite time.