Quasilinear parabolic equations with nonlinear monotone boundary conditions

Chin-Yuan Lin

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 $.

Keywords


m-dissipative operators; method of lines

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