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Nonnegative solutions of a fourth-order nonlinear degenerate parabolic equation. (English) Zbl 0827.35065
The authors study nonnegative solutions of the equation $$u_t + (u^nu_{xxx})_x = 0$$, where $$n$$ is a real positive constant; the equation is degenerated at points $$x$$ at which $$u$$ vanishes. For $$0 < n < 1/2$$, it is shown that initially strictly positive solutions may vanish at some point $$x_0$$ after a finite time $$t_0$$. This result is of particular interest for several hydrodynamic applications in which the above equation arises. For example, if $$u$$ is interpreted as the thickness of a liquid film, it implies that at time $$t_0$$ the film breaks at $$x_0$$.
Reviewer: O.Titow (Berlin)

MSC:
 35K65 Degenerate parabolic equations 35Q35 PDEs in connection with fluid mechanics
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References:
 [1] F. Bernis, Nonlinear parabolic equations arising in semiconductor and viscous droplets models, W.-M. Ni, L. A. Peletier & J. Serrin, editors, Birkhäuser, Boston (1992), 77-88. · Zbl 0793.35107 [2] F. Bernis & A. Friedman, Higher order nonlinear degenerate parabolic equations, J. Differential Equations 83 (1990), 179-206. · Zbl 0702.35143 · doi:10.1016/0022-0396(90)90074-Y [3] F. Bernis, L. A. Peletier & S. M. Williams, Source type solutions of a fourth order nonlinear degenerate parabolic equation, Nonlinear Analysis T. M. A. 18 (1992), 217-233. · Zbl 0778.35056 · doi:10.1016/0362-546X(92)90060-R [4] A. L. Bertozzi, M. P. Brenner, T. F. Dupont & L. P. Kadanoff, Singularities and similarities in interface flows, Trends and perspectives in Applied Mathematics. L. Sirovich, editor, Springer-Verlag, Berlin (1994), 155-208. · Zbl 0808.76022 [5] S. Boatto, L. P. Kadanoff & P. Olla, Travelling wave solutions to thin film equations, Phys. Rev. E 48 (1993), 4423-4431. · doi:10.1103/PhysRevE.48.4423 [6] A. A. Lacey, The motion with slip of a thin viscous droplet over a solid surface, Stud. Appl. Math. 67 (1982) 217-230. · Zbl 0505.76112 [7] S. H. Davis, E. Dibenedetto & D. J. Diller, Some a-priori estimates for a singular evolution equation in thin film dynamics, preprint. · Zbl 0854.35014 [8] M. B. Williams & S. H. Davis, Nonlinear theory of film rupture, Journal of Colloid and Interface Science 90 (1982), 220-228. · doi:10.1016/0021-9797(82)90415-5 [9] F. Bernis, Viscous flows, fourth order nonlinear degenerate parabolic equations and singular elliptic problems, to appear in Free Boundary Problems, 1993 Toledo, Diaz, Herrero, Linan & Vazquez, editors, Pitman Research Notes in Mathematics. [10] A. L. Bertozzi, Loss and gain of regularity in a lubrification equation for thin viscous films, to appear in Free Boundary Problems, 1993 Toledo, Diaz, Herrero, Linan & Vazquez, editors, Pitman Research Notes in Mathematics. [11] A. L. Bertozzi & M. Pugh, The lubrification approximation for thin viscous films: regularity and long time behavior of weak solutions, preprint. · Zbl 0863.76017 [12] A. L. Bertozzi & M. Pugh, The lubrification approximation for thin viscous films: the moving contact line with a porous media cut off of the van der Waals interactions, preprint. · Zbl 0811.35045
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