Math. Ann. (2010) 348:779–795 DOI 10.1007/s00208-010-0496-4 Mathematische Annalen Large time behaviour of solutions to a class of non-autonomous, degenerate parabolic equations F. Ragnedda · S. Vernier Piro · V. Vespri Received: 14 October 2008 / Revised: 21 December 2009 / Published online: 27 February 2010 © Springer-Verlag 2010 Abstract We consider a class of non-autonomous, degenerate parabolic equations and we study the asymptotic behaviour of the solutions. Even if the equation depends explicitly upon the time, we prove that several asymptotic properties, valid for the autonomous case, are preserved in this more general situation. To our knowledge, it is the first time that the asymptotic behaviour of solutions to non-autonomous equations is studied. Mathematics Subject Classification (2000) 35K55 · 35B40 · 35K65 1 Introduction Let  be a bounded domain in R N with C 1 boundary. We consider in  × (t > 0), the following initial boundary value problem u t = div A(x , t , u , ∇u ), (x , t ) ∈  × (t > 0), (1.1) u (x , t ) = 0, (x , t ) ∈ ∂ × (t > 0), (1.2) u (x , 0) = u 0 (x ) ≥ 0, x ∈ , (1.3) F. Ragnedda (B ) · S. Vernier Piro Dipartimento di Matematica e Informatica, Università di Cagliari, Viale Merello 92, 09123 Cagliari, Italy e-mail: ragnedda@unica.it S. Vernier Piro e-mail: svernier@unica.it V. Vespri Dipartimento di Matematica “U. Dini”, Università di Firenze, viale Morgagni 67/a, 50134 Florence, Italy e-mail: vespri@math.unifi.it 123