LOCAL BEHAVIOUR OF SOLUTIONS TO DOUBLY NONLINEAR PARABOLIC EQUATIONS JUHA KINNUNEN AND TUOMO KUUSI Abstract. We give a relatively simple and transparent proof for Harnack’s inequality for certain degenerate doubly nonlinear par- abolic equations. We consider the case where the Lebesgue mea- sure is replaced with a doubling Borel measure which supports a Poincar´ e inequality. 1. Introduction Our purpose is to study the local behaviour of nonnegative weak solu- tions to the doubly nonlinear parabolic equation div(|Du| p−2 Du)= ∂ (u p−1 ) ∂t , 1 <p< ∞. (1.1) When p = 2 we have the standard heat equation. Observe that the solutions to (1.1) can be scaled by nonnegative factors, but due to the nonlinearity of the term (u p−1 ) t we cannot add a constant to a solution. As far as we know, equation (1.1) has first been studied by Trudinger in [Tru], where he proved a Harnack inequality for nonnegative weak solutions. The proof was based on Moser’s celebrated work [Mo1] and used a parabolic version of the John-Nirenberg lemma. Twenty years later the proof of the parabolic John-Nirenberg lemma was simplified by Fabes and Garofalo, see [FaGa]. However, the parabolic BMO re- mains to be technically demanding. Our main objective is to give a relatively simple and transparent proof for Harnack’s inequality using the approach of Moser in [Mo2]. In particular, the parabolic John- Nirenberg lemma is replaced with a lemma due to Bombieri in [BoGi] and [Bomb]. Let us point out a slightly unexpected phenomenon re- lated to the parabolic BMO. In the case p = 2 it is known that if u is a nonnegative solution, then log u is a subsolution to the same equation. However, if p = 2, then log u is not a subsolution to equation (1.1). In- stead it is a subsolution to an equation of the p-parabolic type studied in [DiBe]. 2000 Mathematics Subject Classification. 35K60. Key words and phrases. Harnack inequality, Moser iteration, p-Laplace equation. 1