Nonlinear diffusions and optimal constants in Sobolev type inequalities: asymptotic behaviour of equations involving the 𝐩 -Laplacian

We study the asymptotic behaviour of nonnegative solutions to: u t = Δ p u m using an entropy estimate based on a sub-family of the Gagliardo–Nirenberg inequalities – or, in the limit case m =( p −1) −1 , on a logarithmic Sobolev inequality in W 1, p – for which optimal functions are known.