A strongly ill-posed problem for a degenerate parabolic equation with unbounded coefficients in an unbounded domain &$Omega times {mathcal {O}}$; of &${mathbb {R}}^{M+N}$;

Author： Lorenzi A Lorenzi L

Publisher： IOP Publishing

ISSN： 0266-5611

Source： Inverse Problems, Vol.29, Iss.2, 2013-02, pp. : 25007-25028

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Abstract

In this paper, we deal with a strongly ill-posed second-order degenerate parabolic problem in the unbounded open set &$Omega times {mathcal {O}}subset {mathbb {R}}^{M+N}$;, related to a linear equation with unbounded coefficients, with no initial condition, but endowed with the usual Dirichlet condition on &$(0,T)times partial (Omega times {mathcal {O}})$; and an additional condition involving the x-normal derivative on &$Gamma times {mathcal {O}}$;, with &Ggr; being an open subset of &OHgr;. The purpose of this paper is twofold: to determine sufficient conditions on our data implying the uniqueness of the solution u to the boundary value problem and determine a pair of metrics with respect to which u depends continuously on the data. The results obtained for the parabolic problem are then applied to a similar problem for a convolution integrodifferential linear parabolic equation.