Asymptotic and other properties of a nonlinear diffusion model (Q1276376)

Let \(\Omega\in \mathbb{R}^3\) be a sufficiently smooth bounded fixed domain ensuring the validity of divergence-like theorems. The authors consider the initial-boundary value problem \[ u_t= \Delta F(u),\quad (x,t)\in\Omega\times \mathbb{R}^+, \] \[ u(x,0)= u_0(x),\quad x\in \Omega,\quad u(x,t)= u_1(x),\quad (x,t)\in \partial\Omega\times \mathbb{R}^+, \] where \(F\in C^2(\mathbb{R})\) and \(u_0\in C^2(\overline\Omega)\), \(u_i\in C(\Omega)\) \((i= 0,1)\) are assigned functions. The authors study matters relating to the asymptotic behaviour of solutions. In particular, the authors are interested in obtaining conditions on \(F\) under which each solution tends toward a steady state when \(t\to\infty\).