Time rescaling and asymptotic behavior of some fourth order degenerate diffusion equations (Q2782944)

In the present article, the authors study the asymptotic behaviour of the fourth-order nonlinear degenerate diffusion equation NEWLINE\[NEWLINE u_t = \nabla_x\cdot (|u^n|\nabla_x\Delta_x u),\quad n > 0,NEWLINE\]NEWLINE modelling several physical processes: the evolution of thin film, the motion of spreading viscous droplets, oxidation of silicon in semiconductor devices and others. The main aim is to obtain explicit rates of exponential decaying \(u(x,t)\) in the \(L_1\)-norm. This is made possible by Csiszar-Kullback-type inequalities, which allow for \(L^1\)-bounds in terms of convex entropies. The starting point of long-time analysis mainly relies on nonlinear rescaling and entropy dissipation techniques. In more detail, it is derived exponential time decay of the solution \(u(x,t)\) to the fourth-order problem under study towards its mean value \(\int_{\Omega}u dx\) for all \(n > 0\). Moreover, for the case \(2 < n < 3\) the authors improve the results by Bertozzi and Pugh. It is used slightly different entropies, which allow for explicit bounds. Hence, the authors do reveal the large-times rates of convergence of solutions of the fourth-order problem toward their mean value for all \(n > 0\) in \(L^1\)-norm and make the exponential dependence on \(n\) explicit.