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

Abstract

We are interested in the following class of equations: ( KSDV ) α γ β { h t = ( h h x x + ( 1 2 − α ) ( h x ) 2 ) x x − γ ( ( h x ) 3 h ) x + β h x ( 6 ) , h ( t ) -periodic on ] − 1 , + 1 [ , t ⩾ 0 , h ( 0 ) = h 0 , with α ∈ R , 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 h 0 ⩾ 0 , γ ⩾ 2 3 α then any admissible weak local solution h is necessarily nonnegative. Moreover, there is no global weak solution on R + of ( KSDV ) α γ 0 and the blow up time must occur before T 0 = 2 h ¯ 0 h 0 2 ¯ − ( h ¯ 0 ) 2 provided that h 0 is nonconstant, h ¯ 0 is the average of h 0 over ] − 1 , + 1 [ . On the other hand, if h 0 ⩽ 0 then we have a value α c ⩽ 5 6 such that if α > α c , for all T > 0 , there is a nonpositive global weak solution h on [ 0 , T ] being in particular in L 2 ( 0 , T ; H per 2 ( ] − 1 , + 1 [ ) ) . And if α ⩽ α c , 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 , h 0 ⩽ 0 , then we show that there exists a global solution h ⩽ 0 provided that h 0 belongs to a certain class of functions. If h 0 ⩾ 0 blow-up should occur in this case. We show this fact under the Dirichlet boundary conditions for the weak solution of ( KSDV ) 00 β . The blow up time T max ⩽ −1 λ 1 3 β ln ( 1 − λ 1 β h 0 φ 1 ¯ ) = t ∗ , assuming h 0 φ 1 ¯ > λ 1 β , lim t → T max ∫ 0 t | h ( σ ) | L 1 ( ] − 1 , + 1 [ ) 6 d σ = + ∞ .