Bounded and periodic solutions of quasilinear parabolic equations in time-dependent domains

Abstract

We show the existence and uniqueness of the bounded or periodic solution for the quasilinear parabolic equation of the form(1.1)$u_t-\text{div}\left(\sigma \right(\left|\nabla u{|}^2\right)\nabla u)=f(x,t)\text{ in }Q(-\infty ,\infty )$ with the boundary condition $u\left(t\right){|}_{\partial \Omega \left(t\right)}=0$, where $\Omega \left(t\right)$ is a bounded domain in $R^N$ for each $t\in R$ and $Q(-\infty ,\infty )={\cup }_{-\infty <t<\infty }\Omega \left(t\right)\times \left\{t\right\}$. Typical examples of σ are $\sigma \left(v^2\right)=|v{|}^m,m\ge 0,\sigma (v^2)=\text{log}(1+v^2)$ and $\sigma \left(v^2\right)=|v{|}^m/\sqrt{1+v^2},m\ge 1$. We derive a precise estimate for ${\sup }_{-\infty <t<\infty }{\Vert \nabla u\left(t\right)\Vert }_{\Omega \left(t\right),\infty }$ depending on ${\sup }_{-\infty <t<\infty }{\Vert f\left(t\right)\Vert }_{\Omega \left(t\right),\infty }$ and the movement of $\partial \Omega \left(t\right)$.