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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 $$u_0$$.

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