Estimates for viscosity solutions of parabolic equations with Dirichlet boundary conditions

Gripenberg, G.

doi:10.1090/S0002-9939-02-06580-2

Estimates for viscosity solutions of parabolic equations with Dirichlet boundary conditions
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by G. Gripenberg

Proc. Amer. Math. Soc. 130 (2002), 3651-3660

DOI: https://doi.org/10.1090/S0002-9939-02-06580-2

Published electronically: May 1, 2002

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Abstract:

It is shown how one can get upper bounds for $|u-v|$ when $u$ and $v$ are the (viscosity) solutions of \begin{equation*} u_t - \alpha (D_x u) \Delta _x u = 0\quad \text {and}\quad v_t - \beta (D_x v) \Delta _x v = 0, \end{equation*} respectively, in $(0,\infty )\times \Omega$ with Dirichlet boundary conditions. Similar results are obtained for some other parabolic equations as well, including certain equations in divergence form.

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