Transversality and separation of zeros in second order differential equations

Laister, R.; Beardmore, R.

doi:10.1090/S0002-9939-02-06546-2

Transversality and separation of zeros in second order differential equations
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by R. Laister and R. E. Beardmore PDF

Proc. Amer. Math. Soc. 131 (2003), 209-218 Request permission

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.

References

Donald Aronson, Michael G. Crandall, and L. A. Peletier, Stabilization of solutions of a degenerate nonlinear diffusion problem, Nonlinear Anal. 6 (1982), no. 10, 1001–1022. MR 678053, DOI 10.1016/0362-546X(82)90072-4
D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, Partial differential equations and related topics (Program, Tulane Univ., New Orleans, La., 1974) Lecture Notes in Math., Vol. 446, Springer, Berlin, 1975, pp. 5–49. MR 0427837
David W. Boyd, Best constants in a class of integral inequalities, Pacific J. Math. 30 (1969), 367–383. MR 249556
R. C. Brown and D. B. Hinton, Opial’s inequality and oscillation of 2nd order equations, Proc. Amer. Math. Soc. 125 (1997), no. 4, 1123–1129. MR 1401728, DOI 10.1090/S0002-9939-97-03907-5
Robert Stephen Cantrell, Chris Cosner, and Vivian Hutson, Ecological models, permanence and spatial heterogeneity, Rocky Mountain J. Math. 26 (1996), no. 1, 1–35. MR 1386150, DOI 10.1216/rmjm/1181072101
M. Delgado and A. Suárez, On the existence of dead cores for degenerate Lotka-Volterra models, Proc. Roy. Soc. Edinburgh Sect. A 130 (2000), no. 4, 743–766. MR 1776674, DOI 10.1017/S0308210500000391
B. Gidas, Wei Ming Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), no. 3, 209–243. MR 544879
Morton E. Gurtin and Richard C. MacCamy, On the diffusion of biological populations, Math. Biosci. 33 (1977), no. 1-2, 35–49. MR 682594, DOI 10.1016/0025-5564(77)90062-1
Man Kam Kwong and Li Qun Zhang, Uniqueness of the positive solution of $\Delta u+f(u)=0$ in an annulus, Differential Integral Equations 4 (1991), no. 3, 583–599. MR 1097920
Michel Langlais and Daniel Phillips, Stabilization of solutions of nonlinear and degenerate evolution equations, Nonlinear Anal. 9 (1985), no. 4, 321–333. MR 783581, DOI 10.1016/0362-546X(85)90057-4
A.M. Lyapunov, Problème général de la stabilité du mouvement, Ann. de la Faculté de Toulouse 9 (1907), no. 2, 406.
D. S. Mitrinović, J. E. Pečarić, and A. M. Fink, Inequalities involving functions and their integrals and derivatives, Mathematics and its Applications (East European Series), vol. 53, Kluwer Academic Publishers Group, Dordrecht, 1991. MR 1190927, DOI 10.1007/978-94-011-3562-7
Z. Opial, Sur une inegalite, Ann. Polon. Math. 8 (1960), 29–32.
K. A. Hirsch, On skew-groups, Proc. London Math. Soc. 45 (1939), 357–368. MR 0000036, DOI 10.1112/plms/s2-45.1.357
J. Smoller and A. Wasserman, Global bifurcation of steady-state solutions, J. Differential Equations 39 (1981), no. 2, 269–290. MR 607786, DOI 10.1016/0022-0396(81)90077-2
Fu Hsiang Wong, Zeros of solutions of a second order nonlinear differential inequality, Proc. Amer. Math. Soc. 111 (1991), no. 2, 497–500. MR 1034889, DOI 10.1090/S0002-9939-1991-1034889-7
Fu Hsiang Wong, Cheh Chih Yeh, and Shiueh Ling Yu, A maximum principle for second order nonlinear differential inequalities and its applications, Appl. Math. Lett. 8 (1995), no. 4, 91–96. MR 1340756, DOI 10.1016/0893-9659(95)00055-U
Eberhard Zeidler, Nonlinear functional analysis and its applications. I, Springer-Verlag, New York, 1986. Fixed-point theorems; Translated from the German by Peter R. Wadsack. MR 816732, DOI 10.1007/978-1-4612-4838-5