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Continuous dependence on the nonlinearity of viscosity solutions of parabolic equations. (English) Zbl 0973.35107

The paper proposes an upper bound for \((u-v)\) where \(u\) is a viscosity subsolution of \( u_t+F(u,D_xu,D^2_{xx}u)=0 \) and \(v\) is a viscosity supersolution of \( v_t+G(v,D_xv,D^2_{xx}v)=0 \) in \((0,\infty)\times\Omega\) with Neumann boundary conditions and where \(\Omega\) is an open convex set.

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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References:

[1] B. Cockburn, D. A. French, and, T. E. Peterson, A posteriori error estimates for general numerical methods for Hamilton-Jacobi equations, Math. Comp, to appear.; B. Cockburn, D. A. French, and, T. E. Peterson, A posteriori error estimates for general numerical methods for Hamilton-Jacobi equations, Math. Comp, to appear. · Zbl 1063.65093
[2] Cockburn, B.; Gripenberg, G., Continuous dependence on the nonlinearities of solutions of degenerate parabolic equations, J. Differential Equations, 151, 231-251 (1999) · Zbl 0921.35017
[3] Crandall, M. G.; Ishii, H.; Lions, P. L., User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27, 1-67 (1992) · Zbl 0755.35015
[4] Crandall, M. G.; Lions, P. L., Two approximations of solutions of Hamilton-Jacobi equations, Math. Comp., 43, 1-19 (1984) · Zbl 0556.65076
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