On equilibrium solutions of a bistable reaction-diffusion equation with nonlocal convection (Q674579)

Stationary solution of a reaction-diffusion equation with nonlocal convection satisfies \[ du''+ \Biggl[\Biggl(\int^x_{-\infty} u(y)dy- \int^\infty_x u(y),dy\Biggr)u\Biggr]'+ ku(1-u)(u- a)=0,\tag{1} \] \(x\in\mathbb{R}\), where \(d>0\), \(k>0\), \(0<a<1\) and \('\) denotes \(d/dx\). A nonnegative solution \(u\) of (1) which belongs to \(C^2(\mathbb{R})\cap L^1(\mathbb{R})\cap L^\infty(\mathbb{R})\) is searched. It is proved that there exists an \(a_0>0\) such that for \(0<a<a_0\) equation (1) has two positive symmetric solutions.