Blow-up with logarithmic nonlinearities
نویسندگان
چکیده
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) log(u + 1) (x, t) ∈ R+ × (0, T ), −ux(0, t) = (u + 1) log(u + 1)(0, t) t ∈ (0, T ), u(x, 0) = u0(x) x ∈ R+, with p, 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 blowup 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. 2000 AMS Subject Classification: 35B35, 35B40, 35K55.
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