## Persistence of wavefronts in delayed nonlocal reaction-diffusion equations.(English)Zbl 1117.35037

This paper is concerned with the persistence of wave fronts of the following non-local delay equation $$\frac{\partial u(x,t)}{\partial t}=D\Delta u(x,t)+F\left( u(x,t),\int_{-\tau }^0\int_\Omega du_\tau (\theta ,y)g(u(x+y,t+\theta ))\right) ,$$ where $$x\in \mathbb{R}^m,$$ $$t\geq 0,$$ $$u(x,t)\in \mathbb{R}^n,$$ $$D=\text{diag}(d_1,\cdots ,d_n)$$ with positive constants $$d_i>0,$$ $$\tau$$ is a positive constant, $$u_\tau$$ is a bounded variation function on $$\left[ -\tau ,0\right] \times \Omega \subseteq \left[ -\tau ,0\right] \times \mathbb{R}^m$$ with values in $$\mathbb{R}^{n\times n}$$ and normalized so that $$\int_{-\tau }^0\int_\Omega du_\tau (\theta ,y)=1,$$ and this measure may be dependent on $$\tau ,$$ $$F: \mathbb{R}^n\times \mathbb{R}^n\rightarrow \mathbb{R}^n$$, and $$g:\mathbb{R}^n\rightarrow \mathbb{R}^n$$ are $$C^2$$-smooth functions.
By the perturbation argument and Fredholm alternative theory, they proved the persistence of traveling wavefronts for the above reaction-diffusion equations with nonlocal and delayed nonlinearities if the time lag is relatively small. Namely, when $$\tau$$ is sufficiently small and the reduced version of an ordinary reaction-diffusion system $$\frac{\partial u(x,t)}{\partial t}=D\Delta u(x,t)+F\left( u(x,t),u(x,t)\right)$$ has a travelling wave front, then the above non-local delay equations admit such a solution too, which is true in both monostable and bistable cases and has no requirement on the monotonicity of $$F$$ and $$g.$$ To illustrate the abstract theory, these results are applied to five famous reaction-diffusion models with non-local delay, including a single species model with age structure, the Fisher model, the spatial spread of rabies by red foxes, a bio-reactor model and a hyperbolic model arising from the slow movement of individuals.

### MSC:

 35K57 Reaction-diffusion equations 35K55 Nonlinear parabolic equations 35R10 Partial functional-differential equations 92D25 Population dynamics (general) 92D30 Epidemiology 92D40 Ecology
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