We describe various blow-up patterns for the fourth-order one-dimensional semilinear parabolic equation \[ u_t = - u_{xxxx} + \b [(u_x)^3]_x + e^{u} \] with a parameter $\b \ge 0$, which is a model equation from explosion-convection theory. Unlike the classical Frank-Kamenetskii equation $u_t=u_{xx} +e^u$ (a solid fuel model), by using analytical and numerical evidence, we show that the generic blow-up in this fourth-order problem is described by a similarity solution $u_*(x,t) = -\ln(T\,{-}\,t) \,{+}\, f_1(x/(T\,{-}\,t)^{1/4})(T>0$ is the blow-up time), with a non-trivial profile $f_1 \not \equiv 0$. Numerical solution of the PDE shows convergence to the self-similar solution with the profile $f_1$ 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.