On pushed wavefronts of monostable equation with unimodal delayed reaction (Q2075112)

This paper investigates the wavefront velocities of the delayed reaction-diffusion equation \[ u_{t}(t,x)=u_{xx}(t,x)-u(t,x)+g(u(t-h,x)). \] In the case of ``small delay'' and unimodal $g$, it is proven -- as expected -- that the set of admissible velocities has the structure of a closed ray and that critical velocity wavefronts decay expontentially as $t\to-\infty$. A linearized version of the equation is analysed as the product of two differential operators, and several useful relations are thus obtained, including a representation of the solution as a convolution with the fundamental solution. It follows an in-depth analysis of a ``toy'' model with \(g(u):=\{ku,0\leq u<1-u+4,1\leq u\}\).