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Transversality and separation of zeros in second order differential equations

Authors: R. Laister and R. E. Beardmore
Journal: Proc. Amer. Math. Soc. 131 (2003), 209-218
MSC (2000): Primary 34C10, 34A12, 34A34; Secondary 34B15, 34B60
Published electronically: May 17, 2002
MathSciNet review: 1929040
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Abstract: Sufficient conditions on the non-linearity $f$ are given which ensure that non-trivial solutions of second order differential equations of the form $Lu=f(t,u)$ have a finite number of transverse zeros in a given finite time interval. We also obtain a priori lower bounds on the separation of zeros of solutions. In particular our results apply to non-Lipschitz non-linearities. Applications to non-linear porous medium equations are considered, yielding information on the existence and strict positivity of equilibrium solutions in some important classes of equations.

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Additional Information

R. Laister
Affiliation: School of Mathematical Sciences, University of the West of England, Frenchay Campus, Bristol, England BS16 1QY

R. E. Beardmore
Affiliation: Department of Mathematics, Imperial College, London, England SW7 2BZ

Keywords: Differential equations, transverse zeros, non-Lipschitz non-linearity, separation of zeros, porous medium equations
Received by editor(s): May 29, 2001
Received by editor(s) in revised form: August 25, 2001
Published electronically: May 17, 2002
Communicated by: Carmen C. Chicone
Article copyright: © Copyright 2002 American Mathematical Society

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