Geodesic
Necessary and sufficient condition for the stabilization of the solution of~a~mixed problem for nondivergence parabolic equations to zero
Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 75 (2014) no. 2, pp. 277-308

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We consider the first boundary value problem in a cylindrical domain for a uniformly parabolic second-order equation in nondivergence form. The solution satisfies the homogeneous Dirichlet condition on the lateral surface of the cylinder, and the initial function is bounded. We show that if the coefficients of the equation satisfy the local and global Dini conditions, then a necessary and sufficient condition for the stabilization of the solution to zero coincides with a similar condition for the heat equation. 
@article{MMO_2014_75_2_a9,
     author = {Yu. A. Alkhutov and V. N. Denisov},
     title = {Necessary and sufficient condition for the stabilization of the solution of~a~mixed problem for nondivergence parabolic equations to zero},
     journal = {Trudy Moskovskogo matemati\v{c}eskogo ob\^{s}estva},
     pages = {277--308},
     publisher = {mathdoc},
     volume = {75},
     number = {2},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MMO_2014_75_2_a9/}
}
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Yu. A. Alkhutov; V. N. Denisov. Necessary and sufficient condition for the stabilization of the solution of~a~mixed problem for nondivergence parabolic equations to zero. Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 75 (2014) no. 2, pp. 277-308. http://geodesic.mathdoc.fr/item/MMO_2014_75_2_a9/