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Traveling wavefronts for time-delayed reaction-diffusion equation. II: Nonlocal nonlinearity. (English) Zbl 1173.35072
The author in the second part of a series of study on the stability of traveling wavefronts of reaction-diffusion equations with time delays consider a nonlocal time-delayed reaction-diffusion equation. When the initial perturbation around the traveling wave decays exponentially as $x\rightarrow -\infty $ (but the initial perturbation can be arbitrarily large in other locations), they prove the asymptotic stability of all traveling waves for the reaction-diffusion equation, including even the slower waves whose speed are close to the critical speed. The approach used herein is the weighted energy method, but the weight function is more tricky to construct due to the property of the critical wavefront, and the difficulty arising from the nonlocal nonlinearity is also overcome. Finally, by using the Crank-Nicholson scheme, they present some numerical results to support their theoretical study. [For part I see ibid., 495--510 (2009; Zbl 1173.35071).]

35K57Reaction-diffusion equations
34K20Stability theory of functional-differential equations
92D25Population dynamics (general)
35R10Partial functional-differential equations
35B35Stability of solutions of PDE
35K15Second order parabolic equations, initial value problems
Full Text: DOI
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