Estimates for Viscosity Solutions of Parabolic Equations with Dirichlet Boundary Conditions

It is shown how one can get upper bounds for ju ? vj when u and v are the (viscosity) solutions of ut ? (Dxu))xu = 0 and vt ? (Dxv))xv = 0; respectively, in (0;1) with Dirichlet boundary conditions. Similar results are obtained for some other parabolic equations as well, including certain equations in divergence form. 1. Introduction In this paper we study the problem of how to estimate the diierence between the (viscosity) solutions u and v of the parabolic equations u t ? (D x u)) x u = 0 and v t ? (D x v)) x v = 0;