On interfaces and oscillatory solutions of higher-order semilinear parabolic equations with non-Lipschitz nonlinearities

We study local properties of solutions and their asymptotic extinction behavior for the fourth-order semilinear parabolic equation of diffusion-absorption type u(t) = - u(xxxx) - vertical bar u vertical bar(p-1)u in R x R+, where p < 1, so that the absorption term is not Lipschitz continuous at u = 0. The Cauchy problem with bounded compactly supported initial data possesses solutions with finite interfaces, and we describe their oscillatory, sign changing properties for p epsilon (-1/3, 1). For p epsilon ( 0, 1), we also study positive solutions of the free-boundary problem with zero contact angle and zero-flux conditions. Finally, we describe families {f(k)} of similarity extinction patterns u(S)( x, t) = (T - t)(1/(1- p)) f (y), where y = x/(T - t)(1/ 4), that vanish in finite time, as t -> T- epsilon ( 0, infinity). Similar local and asymptotic properties are indicated for the sixth-order equation with source u(t) = u(xxxxxx) +/- vertical bar u vertical bar(p-1)u, where p epsilon (-1/5, 1).