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Abstract: We give sufficient conditions for the existence of positive travelling wavesolutions for multi-dimensional autonomous reaction-diffusion systems withdistributed delay. To prove the existence of travelling waves, we give anabstract formulation of the equation for the wave profiles in some suitableBanach spaces, and apply known results about the index of some associatedFredholm operators. After a Liapunov-Schmidt reduction, these waves areobtained via the Banach contraction principle, as perturbations of a positiveheteroclinic solution for the associated system without diffusion, whoseexistence is proven under some requirements. By a careful analysis of theexponential decay of the travelling wave profiles at $-\infty$, theirpositiveness is deduced. The existence of positive travelling waves isimportant in terms of applications to biological models. Our method applies tosystems of delayed reaction-diffusion equations whose nonlinearities are notrequired to satisfy a quasi-monotonicity condition. Applications are given, andinclude the delayed Fisher-KPP equation.



Author: Teresa Faria, Sergei Trofimchuk

Source: https://arxiv.org/



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