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Title: Generalized Viscosity Solutions of Fully Non-linear Parabolic Equations with Distributions as Initial Conditions
Authors: IFIDON, E.O
Issue Date: 2005
Publisher: Published in European Journal of Scientific Research
Series/Report no.: 8;1
Abstract: Consider the fully nonlinear parabolic problem   u (D u,Du,u,x,t) 0 Q 0,T T 0 2 t       f   u u (x) Q 0 2 10    where                         i j i 2 2 t x u, Du x x u, D u t uu .  is a bounded open set in Rn and  Zn is a positive integer. If u (x) 0 is a distribution with compact support, it is well known that the classical theory for viscosity solutions does not cover the case where f is discontinuous. This is because the straight forward method of comparing sub and super solutions does not work if f is discontinuous with respect to x and t. We obtain existence and uniqueness results for this class of problems by introducing the concept of Generalized Viscosity Solutions valid in the general case in which the equations are elements of the space of Generalized Functions. No linearization of the equations is assumed, that is our theory is fully non-linear. The solutions thus obtained are shown to be consistent with the classical distributional solutions whenever they exist.
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