Travelling fronts in the diffusive Nicholson's blowflies equation with distributed delays (Q1591885)

The author considers some equations of the form \[ \partial u/\partial t= \partial^2u/\partial x^2- u+\beta(f* u) e^{-(f* u)}, \] where \((f* u)(x, t)= \int^t_{-\infty} f(t- s)u(x, s) ds\) (the kernel \(f: [0,\infty)\to [0,\infty)\) satisfies: \(f(t)\geq 0\), \(\forall t\geq 0\) and \(\int^\infty_0 f(t) dt= 1\)) and \(\beta> 1\) is a parameter. He seeks travelling wave front solutions \(u(x, t)= U(z)\), \(z= x-ct\), \(c>0\), in connection with the steady state solutions \(u=0\) and \(u= \ln\beta\). The existence of such travelling solutions is proved when \(f(t)\) assumes a special form; some qualitative properties of these solutions are established.