Gripenberg, G. Estimates for viscosity solutions of parabolic equations with Dirichlet boundary conditions. (English) Zbl 1090.35097 Proc. Am. Math. Soc. 130, No. 12, 3651-3660 (2002). Summary: It is shown how one can get upper bounds for \(| u-v |\) when \(u\) and \(v\) are the (viscosity) solutions of \[ u_t - \alpha(D_x u) \Delta_x u = 0 \text{ and } v_t - \beta(D_x v) \Delta_x v = 0, \] 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. Cited in 3 Documents MSC: 35K55 Nonlinear parabolic equations 35K65 Degenerate parabolic equations 35K20 Initial-boundary value problems for second-order parabolic equations Keywords:viscosity solution; parabolic; dependence on data; Dirichlet boundary condition PDFBibTeX XMLCite \textit{G. Gripenberg}, Proc. Am. Math. Soc. 130, No. 12, 3651--3660 (2002; Zbl 1090.35097) Full Text: DOI References: [1] B. Cockburn, G. Gripenberg, and S-O. Londen. Continuous dependence on the nonlinearity of viscosity solutions of parabolic equations. J. Differential Equations, 170:180-187, 2001. CMP 2001:08 · Zbl 0973.35107 [2] Michael G. Crandall, Viscosity solutions: a primer, Viscosity solutions and applications (Montecatini Terme, 1995) Lecture Notes in Math., vol. 1660, Springer, Berlin, 1997, pp. 1 – 43. · Zbl 0901.49026 · doi:10.1007/BFb0094294 [3] Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 1 – 67. · Zbl 0755.35015 [4] Diana Nunziante, Existence and uniqueness of unbounded viscosity solutions of parabolic equations with discontinuous time-dependence, Nonlinear Anal. 18 (1992), no. 11, 1033 – 1062. · Zbl 0782.35037 · doi:10.1016/0362-546X(92)90194-J [5] R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. · Zbl 0193.18401 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.