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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Approximate solutions to first and second order quasilinear evolution equations via nonlinear viscosity


Authors: Juan R. Esteban and Pierangelo Marcati
Journal: Trans. Amer. Math. Soc. 342 (1994), 501-521
MSC: Primary 35A35; Secondary 35K65, 35L45
DOI: https://doi.org/10.1090/S0002-9947-1994-1214784-8
MathSciNet review: 1214784
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Abstract: We shall consider a model problem for the fully nonlinear parabolic equation

$\displaystyle {u_t} + F(x,t,u,Du,\varepsilon {D^2}u) = 0$

and we study both the approximating degenerate second order problem and the related first order equation, obtained by the limit as $ \varepsilon \to 0$. The strong convergence of the gradients is provided by semiconcavity unilateral bounds and by the supremum bounds of the gradients. In this way we find solutions in the class of viscosity solutions of Crandall and Lions.

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DOI: https://doi.org/10.1090/S0002-9947-1994-1214784-8
Article copyright: © Copyright 1994 American Mathematical Society