INCOMPLETE BLOW-UP AND SINGULAR INTERFACES FOR QUASI-LINEAR HEAT-EQUATIONS

We study local and asymptotic properties of singular interfaces of the {u = infinity}-level propagation of the blow-up solutions to the quas ilinear heat equation u(t) = (u(m))(xx) + u(p), 0 < m < 1, p = 2 - m. The exponents are chosen in such a particular way because it represent s a limit case of the so-called incomplete blow-up where solutions can be continued after blow-up. We consider the proper solutions of the C auchy problem. We derive the dynamical interface equation and prove th at in general singular interfaces are not analytic in time; they are C -1,C-1 and not C-2-functions. The methods apply to other quasilinear e quations, for instance to the equation with the p-Laplacian operator w hich in the critical incomplete blowup case has the form u(t) = (\u(x) \(sigma)u(x))(x) + u(1/(1+sigma)), - 1 < sigma < 0. We discuss some pr operties of the blow-up interfaces for equations with general nonlinea rities and of the corresponding N-dimensional equations.