### N I Kavallaris, A A Lacey, C V Nikolopoulos and C Voong

#### Abstract

A non-local parabolic equation modelling linear friction welding is studied. The equation applies on the half line and contains a nonlinearity of the form $f(u) \left/ \left( \int_0^\infty f(u) dy \right)^{1 + a} \right.$. For $f(u) = e^u$, global existence and convergence to a steady state are proved. Numerical calculations are also carried out for this case and for $f(u) = (-u)^{1/a}$.