POSITIVE SOLUTIONS FOR A SECOND ORDER
EXTENDED FISHER-KOLMOGOROV'S EQUATION
Diko Souroujon1, Stepan Tersian2 1 University of Economics, Varna
Department of Mathematics
Kniaz Boris I Bvd. 77, Varna – 9002, BULGARIA 2 Institute of Mathematics and Informatics
Bulgarian Academy of Sciences
Acad. G. Bonchev Str., Block 8, Sofia – 1113, BULGARIA
We consider the existence of positive solutions of extended Fisher-Kolmogorov second order differential equation. Using a variational method and an approach of Verzini, we obtain the positive bounded solutions of this ODE.
You will need Adobe Acrobat reader. For more information and free download of the reader, please follow this link.
References
[1] M. Arias, J. Campos, A.M. Robles-Perez, L. Sanchez, Fast and heteroclinic
solutions for a second order ODE related to Fisher-Kolmogorov’s equation.
Calculus of Variations, 21 (2004), 319-334.
[2] D.G. Aronson, H.F. Weinberger, Nonlinear diffusion in population genetics,
combustion and nerve pulse propagation, In: Partial Differential Equations and Related Topics, Lecture Notes in Math., 446, Springer, New York
(1975), 5-49.
[3] A.N. Kolmogorov, I.G. Petrovsky, N.S. Piskunov. ´Etude de l’´equation de
la diffusion avec croissancede la quantite de matiere et son application a
une probl`eme biologique, Bull. Univ. d’Etat a Moscou, Ser. Internationale
A, 1 (1937), 1-26.
[4] Y. Li, Z.Q. Wang, Gluing approximate solutions of minimum type on the
Nehari manifold, In: Proc. USA-Chile Workshop on Nonlinear Analysis
(Via del Mar-Valparaiso, 2000), Electron. J. Diff. Eqns Conf., 6 (2000),
215-223.
[5] C.Marcelli, F. Papalini, Heteroclinic connections for fully nonlinear nonautonomous
second order differential equations, J. Differential Equations,
241 (2007), 160-183.
[6] J. Mawhin, Critical Point Theory and Applications to Nonlinear Differential Equation, Lectures given at the VIGRE Minicourse on Varitional
Method and Nonlinear PDE, University of Utah, May 28-June 8, 2002.
[7] Z. Nehari, Characteristic values associated with a class of nonlinear second
order differential equations, Acta Math., 105 (1961), 141-175.
[8] A. Szulkin, The metod of Nehari manifold revisited (Progress in variational
problems: New trends of geometric gradient flow and critical point theory),
Kyoto University Research Information Repository (2011), 89-102.
[9] G. Verzini, Bounded solutions to superlinear ODEs: a variational approach,
Nonlinearity, 16 (2003), 2013-2028.
[10] V.A. Volpert, Y.M. Suhov, Stationary solutions of non-autonomous
Kolmogorov-Petrovsky-Piskunov equations, Ergodic Theory Dynam. System, 19 (1999).