A nonlocal parabolic problem arising in linear friction welding (Q964009)

The following nonlocal parabolic initial-boundary value problem is considered \[ u_t= u_{xx}-\frac{u^{-p}}{(\int_0^\infty u^{-p}(x,t) \,dx)^{1+1/p}}, \quad 0<x<\infty, \;t>0 \] \[ u_x(0,t)=0, \quad u_x(\infty,t)=1, \quad t>0 \] \[ u(x,0)=u_0(x), \quad x\geq 0 \] where \(p>0\) is a real parameter and \(u_0\) is a positive smooth function. The author proves the existence and uniqueness of the steady state solution for every \(p>1\) (the case physically relevant) and the non-existence for every \(p\in (0,1]\). Moreover, the structure of the steady states is also investigated.