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Asymptotic behaviour of the doubly nonlinear equation $u_t=Δ_p u^m$ on bounded domains

We study the homogeneous Dirichlet problem for the doubly nonlinear equation $u_t = Δ_p u^m$, where $p>1,\ m>0$ posed in a bounded domain in $\mathbb{R}^N$ with homogeneous boundary conditions and with non-negative and integrable data. In this paper we consider the degenerate case $m(p-1)>1$ and the quasilinear case $m(p-1)=1$. We establish the large-time behaviour by proving the uniform convergence to a unique asymptotic profile and we also give rates for this convergence.