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Uniqueness of fast travelling fronts in reaction-diffusion equations with delay. (English) Zbl 1152.35403
Summary: We consider positive travelling fronts, \(u(t,x)=\phi (\nu \cdot x+ct), \phi ( - \infty)=0, \phi (\infty )=\kappa \), of the equation \(u_t(t,x)=\Delta u(t,x) - u(t,x) + g(u(t - h,x)), x\in \mathbb R^m\). This equation is assumed to have exactly two non-negative equilibria: \(u_{1}\equiv 0\) and \(u_{2}\equiv \kappa >0\), but the birth function \(g\in C^{2}(\mathbb R, \mathbb R)\) may be non-monotone on \([0,\kappa ]\). We are therefore interested in the so-called monostable case of the time-delayed reaction-diffusion equation. Our main result shows that for every fixed and sufficiently large velocity \(c\), the positive travelling front \(\phi (\nu \cdot x+ct)\) is unique (modulo translations). Note that \(\phi \) may be non-monotone. To prove uniqueness, we introduce a small parameter \(\varepsilon =1/c\) and realize a Lyapunov-Schmidt reduction in a scale of Banach spaces.

MSC:
35K57 Reaction-diffusion equations
92D25 Population dynamics (general)
35K25 Higher-order parabolic equations
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