Higher order equations related to thin films: Blow-up and global existence, the influence of the initial data

We are interested in the following class of equations: with , γ 0, β 0. When β=0, the model was established by J.R. King [J.R. King, Two generalizations of the thin film equation, Math. Comput. Modelling 34 (2001) 737–756]. Here, we show that if the initial data h0 0, then any admissible weak local solution h is necessarily nonnegative. Moreover, there is no global weak solution on of (KSDV)αγ0 and the blow up time must occur before provided that h0 is nonconstant, is the average of h0 over ]−1,+1[. On the other hand, if h0 0 then we have a value such that if α>αc, for all T>0, there is a nonpositive global weak solution h on [0,T] being in particular in . And if α αc, we show that there exists a weak solution, if γ is greater than (1−α)2. Moreover, adapting the energy method used by Bernis [F. Bernis, Finite speed of propagation and continuity of the interface for thin viscous flows, Adv. Differential Equations 1 (3) (1996) 337–368] in the case α=1 and γ=β=0, we can show that weak solutions have a finite speed of propagation. When β 0, α=γ=0, the model was established by Spencer, Davis and Voorhees [B.J. Spencer, S.H. Davis, P.W. Voorhees, Morphological instability in epitaxially-strained dislocation-free solid films: Nonlinear evolution, Appl. Math. Technical report 9201, Dept. of Engineering Sci. and Appl. Math., McCormick School of Eng. and Appl. Sci. Northwestern, University Evanston, IL 60208 (September 1992)]. If β>0, h0 0, then we show that there exists a global solution h 0 provided that h0 belongs to a certain class of functions. If h0 0 blow-up should occur in this case. We show this fact under the Dirichlet boundary conditions for the weak solution of (KSDV)00β. The blow up time , assuming , .