On the free boundary for quenching type parabolic problems via local energy methods (Q479328)

Under consideration is the parabolic problem \[ \frac{\partial \psi(u)}{\partial t} - \text{div\,}A(x,t,u,\nabla u) +c(x,t,u)=f(x,t,u), \;\;u(x,0)=u_{0}(x), \;u|_{S}=\varphi(x), \tag{1} \] where \((x,t)\in Q=\Omega\times (0,T)\), \(\Omega\) is a bounded domain in \({\mathbb R}^{n}\), and \(S=\partial\Omega\times (0,T)\). The main assumptions are the inequalities \[ |A(x,t,u,\vec{q})|\leq c_{1}|\vec{q}|^{p-1},\quad c_{2}|\vec{q}|^{p}\leq A(x,t,u,\vec{q})\cdot \vec{q},\quad f(x,t,u)u\leq \lambda |u|^{q+1}+g(x,t)u, \] \[ c_{3}|u|^{\gamma+1}\leq G(u)\leq c_{4}|u|^{\gamma+1}\;(G(u)=\psi(u)u-\int_{0}^{u}\psi(\tau)\,d\tau),\quad c_{5}|u|^{\alpha}\leq c(x,t,u)u, \; \] where \(p>1\), \(q\in {\mathbb R}\), \(\gamma \in (0,p-1]\), \(\alpha\in (0,\min(1, \gamma p/(p-1)))\), \(c_{i}\) are some positive constants, and \(\lambda\in {\mathbb R}\). Under these assumptions, the property of finite speed of propagation of disturbances for equation (1) is established. To some extent, the structure of the set \(\{(x,t):\;u(x,t)=0\}\) is described.