Unbounded solutions to autonomous quasilinear parabolic equations (Q1358040)

Initial-boundary value problems for quasilinear parabolic equations are studied. Besides standard regularity, ellipticity, and growth conditions, it is assumed that zero is a stable equilibrium and there is no positive equilibrium. It is then proved that for any global solution \(u(x,t)\) starting from a positive initial datum the following property holds: for any \(x\) the (finite or infinite) limit \(\phi(x)=\lim_{t\to\infty} u(x,t)\) exists and, where finite, \(\phi\) solves the corresponding stationary problem.