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Traveling fronts in monostable equations with nonlocal delayed effects. (English) Zbl 1141.35058
Summary: We study the existence, uniqueness and stability of traveling wave fronts in the following nonlocal reaction-diffusion equation with delay \[ \frac{\partial u\left(x, t\right)}{\partial t}= d\Delta u\left(x, t\right)+f\left(u\left(x, t\right),\int\limits_{-\infty}^\infty h\left(x - y\right) u\left(y, t - \tau\right) dy\right). \] Under the monostable assumption, we show that there exists a minimal wave speed \(c^* > 0\), such that the equation has no traveling wave front for \(0 < c < c^*\) and a traveling wave front for each \(c \geq c^*\). Furthermore, we show that for \(c > c^*\), such a traveling wave front is unique up to translation and is globally asymptotically stable. When applied to some population models, these results cover, complement and/or improve a number of existing ones. In particular, our results show that
(i) if \(\partial_{2} f(0, 0) > 0\), then the delay can slow the spreading speed of the wave fronts and the nonlocality can increase the spreading speed; and
(ii) if \(\partial_{2} f(0, 0) = 0\), then the delay and nonlocality do not affect the spreading speed.

MSC:
35R10 Functional partial differential equations
35K57 Reaction-diffusion equations
35B40 Asymptotic behavior of solutions to PDEs
34K30 Functional-differential equations in abstract spaces
58D25 Equations in function spaces; evolution equations
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