On the rate of drying of gelatin in a photographic film (Q1924503)

It is considered the diffusion of water in a gelatin emulsion. The following has been proposed as a model for this phenomenon: \[ u_t=\text{div}(G(u)Du),\quad (x,t)\in\Omega\times(0,\infty), \] \[ \exp(-u^{-1})u_\eta+u=0\quad\text{on }\Gamma,\quad u_\eta=0\quad\text{on }\partial\Omega\backslash\Gamma,\quad u(x,0)=u_0(x). \] Here, \(u\) represents the volume fraction of water, \(x\) is the Lagrangian coordinate attached to the gelatin, \(G(u)=\exp(-u^{-1})/(1+u)^2\), \(\Omega\subset\mathbb{R}^N\), and \(\Gamma\) is a simply connected relatively open subset of \(\partial\Omega\). The main result of this paper is that any solution of this problem satisfies \[ \sup_\Omega u(\cdot,t)\leq[1/\ln t][1+C(1+M)^{N/r}|u_0|_r/\ln t+4r\ln(\ln t)/(N\ln t)] \] for \(t>\max\{C^{2r/(N+2)},t_*\}\). Moreover, for every \(x_0\in\Omega\), \[ u(x_0,t)\geq 1/[1+\ln(\gamma/(\text{dist}\{x_0,\partial\Omega\})^2)+\ln t]\quad\text{for all}\quad t>\gamma^{(2r-N)/N}|\exp(u^{-1}_0)|^{2r/N}_{r,\Omega}. \]