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Heteroclinic orbits, mobility parameters and stability for thin film type equations. (English) Zbl 1029.35121
In this paper the authors investigate stability of and heteroclinic connections between equilibria of the thin film equation, \[ h_t = -(h^n h_{xxx})_x - {\mathcal B}(h^m h_x)_x, \tag{1} \] where \(n,m >0\) are real numbers, and the Bond number \({\mathcal B}\) is positive.
The study is numerical, using an adaptive time-stepping method described in Appendix A. Four types of equilibria, constant and non-constant periodic, as well as two types of droplets, with zero and non-zero contact angles are considered. It is best to read this paper in conjunction with other work by the authors [e.g. J. Differ. Equ. 182, 377-415 (2002; Zbl 1011.34005)].
Clearly, equilibria and their stability depend on the \({\mathcal B}\) and the difference of \(m\) and \(n\), or, equivalently, on the parameter \(q=m-n+1\). Hence the following questions arise: 1. how does the structure of the set of equilibria, their stability and heteroclinic connections depend on \(q\); 2. how are heteroclinic connections changed by changing, for fixed \(q\), the exponents \(m\) and \(n\); 3. can such a change in \(m\) and \(n\) result in a change of the nature of singularities of equilibria (e.g. the number of film rupture points).
A wealth of very suggestive results is presented. For example, it is found that changing \(n\) and \(m\) does not cause a heteroclinic connection to be broken, but the effects of changing this parameters on the location and number of touchdown points can be very spectacular. See for example the situation in which \(q=5/2\), and the evolution of the same initial data is followed for \(n=1\) and \(n=2\). In the first case there is a unique local minimum of the solution for all time, while in the second case it splits into two. The results indicate the presence of a critical exponent \(n_1(q)\) that governs the number of local minima per period.

35K55 Nonlinear parabolic equations
37C29 Homoclinic and heteroclinic orbits for dynamical systems
37L15 Stability problems for infinite-dimensional dissipative dynamical systems
35B33 Critical exponents in context of PDEs
74K35 Thin films
76D08 Lubrication theory
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