A maximum principle for fully nonlinear parabolic equations with time degeneracy (Q2708575)

Let \(\Omega\subset \mathbb{R}^n\) be a bounded domain with boundary \(\partial_1\Omega\cup\partial_2\Omega\). For \(T>0\), let \(\overline Q_T=\overline\Omega \times [0,T]\), \(Q_T=(\Omega\cup \partial_2\Omega)\times (0,T]\) and \(q_T=\overline Q_T\setminus Q_T\). The fully nonlinear parabolic equation \(b(u)_t=F(x,t,u,\nabla u,D^2u)=: F[u]\) is investigated. On \(\partial_1\Omega \times(0,T]\), inhomogeneous Dirichlet conditions are prescribed, while, on \(\partial_2\Omega \times(0,T]\) the dynamical boundary condition \(B_o(u):= \sigma(x,t)\partial_tu+c(x,t)\partial_ \nu u=0\) is imposed. Here \(c>0\) and \(\sigma\geq 0\).NEWLINENEWLINENEWLINEThe first result is the following. Suppose there are constants \(\beta\geq 0\), \(z_0\geq 1\) such that \(zF(.,.,z,0,0)\leq (\beta z^2+\beta)b'(z)\) and \(b'(z)>0\) for \(|z|>z_0\), \(b(-z_0) =\min_{[-z_0,z_0]}\) and \(b(z_0)=\max_{[-z_0,z_0]}\). Then, any solution \(u\) of the problem \(b(u)_t=F[u]\) in \(Q_T\), \(B_o(u)=0\) on \(\partial_2\Omega\times(0,T]\) satisfies \(\max_{\overline Q_T}|u|\leq e^{2\beta T}\max\Bigl\{\max_{q_T}|u|,z_0\Bigr\}\).NEWLINENEWLINENEWLINEBy a counterexample, it is shown that a comparison principle with respect to \(q_T\) cannot hold. However, the following result is carried out. Let \(u\) and \(v\) satisfy \(B_o(u)=B_o(v)=0\) on \(\partial_2\Omega\times(0,T]\) and the inequality \(b(u)_t-F[u]<b(v)_t-F[v]\) in \(Q_T\). Provided \(b'\geq 0\), if \(u<v\) on \(q_T\) then \(u<v\) in \(Q_T\). Finally, if \(b\) has at most a critical point, a weak maximum principle is proved.NEWLINENEWLINEFor the entire collection see [Zbl 0957.00037].