Large time asymptotics of the doubly nonlinear equation in the non-displacement convexity regime

We study the long-time asymptotics of the doubly nonlinear diffusion equation ${\rho }_t=div(\mathrm{\mid }\mathrm{\nabla }{\rho }^m{\mathrm{\mid }}^{p-2}\mathrm{\nabla }{\rho }^m)$ in ${\text{RR}}^n$, in the range $\frac{n-p}{n(p-1)}<m>\frac{n-p+1}{n(p-1)}$ and $1p\mathrm{\infty }$ where the mass of the solution is conserved, but the associated energy functional is not displacement convex. Using a linearisation of the equation, we prove an $L^1$-algebraic decay of the non-negative solution to a Barenblatt-type solution, and we estimate its rate of convergence. We then derive the nonlinear stability of the solution by means of some comparison method between the nonlinear equation and its linearisation. Our results cover the exponent interval $\frac{2n}{n+1}p\frac{2n+1}{n+1}$ where a rate of convergence towards self-similarity was still unknown for the $p$-Laplacian equation.