Abstract :
We study the existence, uniqueness and asymptotic behavior, as well as the stability of a special kind of traveling wave solutions
for competitive PDE systems involving intrinsic growth, competition, crowding effects and diffusion. The traveling waves are
exclusive in the sense that as the variable goes to positive or negative infinity, different species are close to extinction or carrying
capacity. We perform an appropriate affine transformation of the traveling wave equations into monotone form and construct
appropriate upper and lower solutions. By this means, we reduce the existence proof to application of well-known theory about
monotone traveling wave systems (cf. [A. Leung, Systems of Nonlinear Partial Differential Equations: Applications to Biology and
Engineering, MIA, Kluwer, Boston, 1989; J.Wu, X. Zou, Traveling wave fronts of reaction–diffusion systems with delay, J. Dynam.
Differential Equations 13 (2001) 651–687] and [I. Volpert, V. Volpert, V. Volpert, Traveling Wave Solutions of Parabolic Systems,
Transl. Math. Monogr., vol. 140, Amer. Math. Soc., Providence, RI, 1994]). Then, by using spectral analysis of the linearization
over the profile, we prove the orbital stability of the traveling wave in some Banach spaces with exponentially weighted norm.
Furthermore, we show that the introduction of some weight is necessary in the sense that, in general, traveling wave solutions with
initial perturbations in the (unweighted) space C0 are unstable (cf. [I. Volpert, V. Volpert, V. Volpert, Traveling Wave Solutions of
Parabolic Systems, Transl. Math. Monogr., vol. 140, Amer. Math. Soc., Providence, RI, 1994] and [D. Henry, Geometric Theory
of Semilinear Parabolic Equations, Lecture Notes in Math., vol. 840, Springer-Verlag, New York, 1981]).
© 2007 Elsevier Inc. All rights reserved
