### Quasilinear parabolic equations with nonlinear monotone boundary conditions

DOI: http://dx.doi.org/10.12775/TMNA.1999.012

#### Abstract

Of concern is the following quasilinear parabolic equation with a

nonlinear monotone boundary condition:

$$

\cases

\displaystyle u_{t} (x, t) = \frac{\partial \alpha (x, u_{x})}{\partial x} + g(x, u),

\quad (x, t) \in (0, 1) \times (0, \infty), \\

(\alpha (0, u_{x}(0, t)), - \alpha (1, u_{x}(1, t))) \in \beta (u(0,

t),

u(1, t)), \\

u(x, 0) = u_{0}(x).

\endcases

\tag $*$

$$

Here $ \beta $ is a maximal monotone graph in $ {\mathbb R} \times {\mathbb R}$,

which contains the origin $(0, 0)$. It is showed

that (*) has a unique strong solution $ u $, with the property that

$$

\sup_{t \in [0, T]}\|u(x, t)\|_{C^{1+ \nu}[0, 1]}

$$

is uniformly bounded for $ 0 < \nu < 1 $ and finite $ T > 0 $.

nonlinear monotone boundary condition:

$$

\cases

\displaystyle u_{t} (x, t) = \frac{\partial \alpha (x, u_{x})}{\partial x} + g(x, u),

\quad (x, t) \in (0, 1) \times (0, \infty), \\

(\alpha (0, u_{x}(0, t)), - \alpha (1, u_{x}(1, t))) \in \beta (u(0,

t),

u(1, t)), \\

u(x, 0) = u_{0}(x).

\endcases

\tag $*$

$$

Here $ \beta $ is a maximal monotone graph in $ {\mathbb R} \times {\mathbb R}$,

which contains the origin $(0, 0)$. It is showed

that (*) has a unique strong solution $ u $, with the property that

$$

\sup_{t \in [0, T]}\|u(x, t)\|_{C^{1+ \nu}[0, 1]}

$$

is uniformly bounded for $ 0 < \nu < 1 $ and finite $ T > 0 $.

#### Keywords

m-dissipative operators; method of lines

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