Geodesic
Formulation and well-posedness of the Cauchy problem for a diffusion equation with discontinuous degenerating coefficients
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 49 (2009) no. 6, pp. 1085-1102 
L. V. Korobenko; V. Zh. Sakbaev. Formulation and well-posedness of the Cauchy problem for a diffusion equation with discontinuous degenerating coefficients. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 49 (2009) no. 6, pp. 1085-1102. http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_6_a11/
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     author = {L. V. Korobenko and V. Zh. Sakbaev},
     title = {Formulation and well-posedness of the {Cauchy} problem for a~diffusion equation with discontinuous degenerating coefficients},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1085--1102},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_6_a11/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

The choice of a differential diffusion operator with discontinuous coefficients that corresponds to a finite flow velocity and a finite concentration is substantiated. For the equation with a uniformly elliptic operator and a nonzero diffusion coefficient, conditions are established for the existence and uniqueness of a solution to the corresponding Cauchy problem. For the diffusion equation with degeneration on a half-line, it is proved that the Cauchy problem with an arbitrary initial condition has a unique solution if and only if there is no flux from the degeneration domain to the ellipticity domain of the operator. Under this condition, a sequence of solutions to regularized problems is proved to converge uniformly to the solution of the degenerate problem in $L_1(R)$ on each interval.

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