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Local existence and \(L^ \infty\)-estimate of weak solutions to a nonlinear degenerate parabolic equation with nonlinear boundary data. (English) Zbl 0849.35061
Let \(\Omega\) be a bounded domain in \(\mathbb{R}^N\) with smooth boundary \(\partial \Omega\), \(\nu\) be the outward unit normal defined on \(\partial\Omega\) and let \(0< m\), \(r< \infty\) be given. Consider the problem \[ (|u|^{m- 1} u)_t= \Delta_r u\quad \text{in } \Omega_T,\;\nabla_r u\cdot \nu= f(u)\quad \text{on } S_T,\tag{1} \]
\[ u(x, 0)= u_0(x),\quad u_0\in L^\infty(\Omega)\cap W^1_{r+ 1}(\Omega), \] where \[ \Delta_r u= \sum^N_{i= 1} (|u_{x_i}|^{r- 1} u_{x_i})_{x_i},\quad \nabla_r u= (|u_{x_1}|^{r- 1} u_{x_1},\dots, |u_{x_N}|^{r- 1} u_{x_N}), \] \(0< T< \infty\), \(\Omega_T= \Omega\times (0, T)\), \(S_T= \partial \Omega\times (0, T)\) and a sufficiently smooth function \(f\) satisfies the following growth condition \(f(u)\text{ sign } u\leq L(|u|^\alpha+ 1)\), \(L\geq 0\), for some \(0\leq \alpha< \infty\).
The main concern of this note is to deal with the following problem: Having known that a weak solution \(u\) of problem (1) is qualitatively bounded, find a quantitative \(L^\infty(\Omega_T)\) estimate of \(u\), depending only upon \(m\), \(r\), \(\Omega\), \(N\), \(T\), \(L\), \(\alpha\) and \(u_0\).

35K65 Degenerate parabolic equations
35B45 A priori estimates in context of PDEs