Explicit exponential decay bounds in quasilinear parabolic problems (Q1291874)

The authors study the initial-Dirichlet boundary value problem for the quasilinear parabolic equation \[ \frac {\partial u}{\partial t} =g(k(t)|\nabla u|^2)\Delta u + f(u) \] for given functions \(g\), \(k\), and \(f\). Here, the boundary condition is \(u=0\) on \(\partial\Omega\times (0,T)\) and the initial function \(u(\cdot,0)\) vanishes on \(\partial\Omega\) for a domain \(\Omega \subset \mathbb R^N\). Assuming appropriate technical conditions on \(g\), \(k\), and \(f\), they show that, if \(N=1\) and \(\alpha\) is a nonnegative parameter, then the function \(\Phi\) defined by \[ \begin{gathered} \Phi(x,t)= e^{2\alpha t}\left[\frac {1}{k(t)} G(k(t)u_x(s,t)^2) + \alpha u^2 + 2F(u)\right], \\ F(s) = \int_0^s f(\xi) d\xi,\;G(s) = \int_0^s g(\xi) d\xi, \end{gathered} \] takes on its maximum value at a point \((x_0,t_0)\) with \(u_x(x_0,t_0) =0\) or \(t_0=0\). When \(f\equiv 0\) or \(k\equiv 1\), the first alternative cannot occur for a suitable choice of \(\alpha\), so the solution must exist for all time, and \(u_x\) decays exponentially in time. They also prove some analogous results for the case \(N>1\), which require \(f\equiv 0\). In this case, \(\Phi\) is given by \[ \Phi(x,t)= e^{2\alpha\beta t}\left[\frac {1}{k(t)} G(k(t)|\nabla u(x,t)|^2) + \alpha u^2 \right] \] for some \(\beta \in (0,1)\) which is related to \(\alpha\).