×

zbMATH — the first resource for mathematics

Higher order nonlinear degenerate parabolic equations. (English) Zbl 0702.35143
The paper deals with a few existence and positivity results for higher order, possibly degenerate, nonlinear parabolic equations. The equations under investigation are of the type \[ \frac{du}{dt}+\frac{\partial}{\partial \kappa}(f(u)\frac{\partial^{2m+1}u}{\partial \kappa^{2m+1}})=0 \] where \(f(u)=| u|^ m\) \(f_ 0(u)\) with \(n\geq 1\) and \(f_ 0(u)>0.\)
Existence of a weak solution is shown for \(m\geq 2\). Furthermore if the initial data is nonnegative, the solution is also nonnegative. For \(n\geq 4\), the authors show that the support of the solution u increases with time t (a weaker form of this property being proved for \(2\leq n<4)\).
Reviewer: D.Blanchard

MSC:
35K65 Degenerate parabolic equations
35K55 Nonlinear parabolic equations
35K25 Higher-order parabolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Eidelman, S.D, Parabolic systems, (1969), North-Holland Amsterdam
[2] Friedman, A, Interior estimates for parabolic systems of partial differential equations, J. math. mech., 7, 393-418, (1958) · Zbl 0082.30402
[3] Greenspan, H.P, On the motion of a small viscous droplet that wets a surface, J. fluid mech., 84, 125-143, (1978) · Zbl 0373.76040
[4] Greenspan, H.P; McCay, B.M, On the wetting of a surface by a very viscous fluid, Stud. appl. math., 64, 95-112, (1981) · Zbl 0474.76099
[5] Hocking, L.M, Sliding and spreading of thin two-dimensional drops, Quart. J. mech. appl. math., 34, 37-55, (1981) · Zbl 0487.76094
[6] King, J.R, ()
[7] {\scJ. R. King}, The isolation oxidation of silicon, SIAM J. Appl. Math., in press. · Zbl 0665.76108
[8] {\scJ. R. King}, The isolation oxidation of silicon: the reaction-controlled case, SIAM J. Appl. Math., in press. · Zbl 0672.76095
[9] Lacey, A.A, The motion with slip of a thin viscous droplet over a solid surface, Stud. appl. math., 67, 217-230, (1982) · Zbl 0505.76112
[10] Lions, J.L, Quelques méthodes de résolution des problèmes aux limites non linéaires, (1969), Dunod Paris · Zbl 0189.40603
[11] Smith, N.F; Hill, J.M, High order nonlinear diffusion, IMA J. appl. math., 40, 73-86, (1988) · Zbl 0694.35091
[12] Solonnikov, V.A; Solonnikov, V.A, On boundary value problems for linear parabolic systems of differential equations of general form, (), 1-184, English translation · Zbl 0961.35114
[13] Tayler, A.B; King, J.R, Free boundaries in semi-conductor fabrication, (), in press
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.