Steady states, global existence and blow-up for fourth-order semilinear parabolic equations of Cahn-Hilliard type

Fourth-order semilinear parabolic equations of the Cahn-Hilliard-type u(r) + Delta(2)u = gamma u +/- Delta(vertical bar u vertical bar(p-1)u) in Omega x R+, are considered in a smooth bounded domain Omega subset of R-N with Navier-type boundary conditions on partial derivative Omega, or Omega = R-N, where p > 1 and gamma are given real parameters. The sign " + " in the "diffusion term" on the right-hand side means the stable case, while " - " reflects the unstable (blow-up) one, with the simplest, so called limit, canonical model for gamma = 0, u(1) + Delta(2)u = +/-Delta(vertical bar u vertical bar(p-1)u) in R-N x R+. The following three main problems are studied: (i) for the unstable model (0.1), with the -Delta(vertical bar u vertical bar(p-1)u), existence and multiplicity of classic steady states in Omega subset of R-N and their global behaviour for large gamma > 0; (ii) for the stable model (0.2), global existence of smooth solutions u(x, t) in R-N x R+ for bounded initial data u(0)(x) in the subcritical case p <= p(center dot) = 1 + 4\(N-2)(+); and (iii) for the unstable model (0.2), a relation between finite time blow-up and structure of regular and singular steady states in the supercritical range. In particular, three distinct families of Type I and II blow-up patterns are introduced in the unstable case.

Steady states, global existence and blow-up for fourth-order semilinear parabolic equations of Cahn-Hilliard type Articles