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Traveling wavefronts for time-delayed reaction-diffusion equation. I: Local nonlinearity. (English) Zbl 1173.35071
A class of time-delayed reaction-diffusion equation with local nonlinearity for the birth rate is studied. For all wavefronts with the speed strictly greater than the critical wave speed, it is proved that these wavefronts are asymptotically stable when the initial perturbation around the traveling waves decays exponentially as $x\rightarrow - \infty $, but the initial perturbation can be arbitrarily large in other locations. This essentially improves the stability results in previous studies. The approach adopted in this paper is the technical weighted energy method and based on the property of the critical wavefronts, the weight function is carefully selected and it plays a key role in proving the stability for any speed strictly greater than the critical wave speed and for an arbitrary time-delay. [For part II see ibid., 511--529 (2009; Zbl 1173.35072).]

MSC:
35K57Reaction-diffusion equations
34K20Stability theory of functional-differential equations
92D25Population dynamics (general)
35R10Partial functional-differential equations
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References:
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