Blow-up in a fourth-order semilinear parabolic equation from explosion-convection theory

We describe various blow-up patterns for the fourth-order one-dimensional semilinear parabolic equation \[ u_t = - uₓₓₓₓ + [(u_x)^3]_x + eᵘ \] with a parameter from a wide variety of initial data. We also construct a countable subset of other, not self-similar, blow-up patterns by using a spectral analysis of an associated linearized operator and matching with similarity solutions of a first-order Hamilton–Jacobi equation.