Global Solvability and Blow up for the Convective Cahn-hilliard Equations with Concave Potentials

We study initial boundary value problems for the convective Cahn-Hilliard equation ∂tu+ ∂ 4 xu+ u∂xu+ ∂ 2 x(|u| u) = 0. It is well-known that without the convective term, the solutions of this equation may blow up in finite time for any p > 0. In contrast to that, we show that the presence of the convective term u∂xu in the Cahn-Hilliard equation prevents blow up at least for 0 < p < 4 9 . We also show that the blowing up solutions still exist if p is large enough (p ≥ 2). The related equations like KolmogorovSivashinsky-Spiegel equation, sixth order convective Cahn-Hilliard equation, are also considered. 1. Intoduction It is well-known that the solutions of the semilinear heat equations with concave potentials ∂tu−∆u− u|u| p = 0 blow up in finite time if p > 0 and the initial energy is negative, see e.g. [1, 17, 21, 20, 28] and references therein. However, it is also established that the presence of the convective terms in the semilinear parabolic equation prevents blow up if the nonlinear source term is not growing very rapidly. For instance, the solutions of the following convective heat equation ∂tu+ u∂xu− ∂ 2 xu− u|u| p = 0 (1.1) in a bounded interval Ω = [−L, L] with Dirichlet boundary conditions exist globally in time if p ≤ 1 and the blowing up solutions occur only if p > 1, see [5, 22, 23, 30], see also [32] for the results on suppressing the blow up by adding the sufficiently large linear convective terms in reaction-diffusion equations. The main aim of the present paper is to study the analogous problems for the following convective Cahn-Hilliard (CH) equation with concave potentials ∂tu+ ∂ 2 x(∂ 2 xu+ u|u| ) + u∂xu = 0 (1.2) in a bounded segment Ω = [−L, L] endowed by periodic boundary conditions. The long-time behavior of solutions of initial boundary value problems for CH and related equations are intensively studied by many authors, see [2, 4, 10, 16, 14, 26, 31] and references therein. For instance, the existence of the blowing up solutions for equation (1.2) without the convective term u∂xu is known for any p > 0, see e.g. [10, 11, 24, 27]. However, based on the reaction-diffusion experience mentioned above, one may expect