Topological Methods in Nonlinear Analysis

Quasilinear parabolic equations with nonlinear monotone boundary conditions

Chin-Yuan Lin

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Abstract

Of concern is the following quasilinear parabolic equation with a nonlinear monotone boundary condition: \begin{equation} \begin{cases} 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). \end{cases} \tag{*} \end{equation} 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 $.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 13, Number 2 (1999), 235-249.

Dates
First available in Project Euclid: 29 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1475178880

Mathematical Reviews number (MathSciNet)
MR1742222

Zentralblatt MATH identifier
0962.35101

Citation

Lin, Chin-Yuan. Quasilinear parabolic equations with nonlinear monotone boundary conditions. Topol. Methods Nonlinear Anal. 13 (1999), no. 2, 235--249. https://projecteuclid.org/euclid.tmna/1475178880


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