Asymptotic behaviour of the doubly nonlinear diffusion equation ut=Δpumut=Δpum on bounded domains

We study the homogeneous Dirichlet problem for the doubly nonlinear diffusion equation ut=Δpumut=Δpum, where p>1,m>0p>1,m>0, posed in a bounded domain in RNRN with homogeneous boundary conditions and with non-negative and integrable initial data. In this paper we consider the degenerate case m(p−1)>1m(p−1)>1 and the quasilinear case m(p−1)=1m(p−1)=1. In the first case we establish the large-time behaviour by proving the uniform convergence to a unique asymptotic profile and we also give rates for this convergence. The difference in the second case is that the asymptotic profile is unique up to a constant factor that we determine.