Nonlinear Analysis 114 (2015) 169–185 Contents lists available at ScienceDirect Nonlinear Analysis journal homepage: www.elsevier.com/locate/na On a nonlinear hyperbolic equation with a bistable reaction term B.H. Gilding a,∗ , R. Kersner b a Department of Mathematics, Faculty of Science, Kuwait University, Kuwait b Department of Informatics, Pollack Mihály Faculty of Engineering, University of Pécs, Hungary article info Article history: Received 1 September 2014 Accepted 29 October 2014 Communicated by Enzo Mitidieri MSC: 35L70 35Q99 80A20 92D25 Keywords: Travelling wave Second-order hyperbolic partial differential equation Nonlinear Existence Uniqueness Wave-speed abstract Nonlinear second-order hyperbolic equations are gaining ground as models in many areas of application, as extensions of parabolic reaction–diffusion equations that might other- wise be used. The theory of travelling-wave solutions of such reaction–diffusion equations is well established. The present paper is concerned with its counterpart for the wider class of equations in the particular case that the reaction term is bistable. Conditions that are nec- essary and sufficient for the existence and uniqueness of these solutions are determined. A combination of traditional ordinary differential equation techniques and an innovatory integral equation approach is employed. © 2014 Elsevier Ltd. All rights reserved. 1. Nonlinear analysis This paper is concerned with the analysis of wavefront solutions of the nonlinear equation ε 2 u tt + g (u)u t = (k(u)u x ) x + f (u), (1.1) in which subscripts denote partial differentiation, ε is a positive parameter that is not necessarily small, and f , g , and k are given functions. Equations of this form arise as mathematical models in many settings. These will be reviewed presently. When k and g are positive, the equation is of second-order hyperbolic type and can be classified as a nonlinear telegraph equation. In the limit ε = 0, this equation reduces to a nonlinear reaction–diffusion equation of second-order parabolic type. In that context, k is the diffusion coefficient and f represents the reaction term. A solution of Eq. (1.1) of the form u = ϕ(η) where η = x − ct (1.2) and c is a constant, is called a travelling-wave solution. The constant c —which may be positive, negative, or zero—is known as the wave-speed.A wavefront solution is a travelling-wave solution that is defined for all η ∈ R, monotonic, and such that ∗ Correspondence to: Department of Mathematics, College of Science, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait. Tel.: +965 24985362. E-mail addresses: gilding@sci.kuniv.edu.kw (B.H. Gilding), kersner@pmmik.pte.hu (R. Kersner). http://dx.doi.org/10.1016/j.na.2014.10.036 0362-546X/© 2014 Elsevier Ltd. All rights reserved.