Geodesic
Initial-boundary-value problem for inhomogeneous degenerate equations of mixed parabolic-hyperbolic type
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential equations. Mathematical physics, Tome 137 (2017), pp. 26-60.

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider initial-boundary-value problems for three classes of inhomogeneous degenerate equations of mixed parabolic-hyperbolic type: mixed-type equations with degenerate hyperbolic part, mixed-type equations with degenerate parabolic part, and mixed-type equations with power degeneration. In each case, we state a criterion of uniqueness of a solution to the problem. We construct solutions as series with respect to the system of eigenfunctions of the corresponding one-dimensional spectral problem. We prove that the uniqueness of the solution and the convergence of the series depend on the ratio of sides of the rectangular from the hyperbolic part of the mixed domain. In the proof of the existence of solutions to the problem, small denominators appear that impair the convergence of series constructed. In this connection, we obtain estimates of small denominators separated from zero and the corresponding asymptotics, which allows us, under certain conditions, to prove that the solution constructed belongs to the class of regular solutions.
Keywords: mixed parabolic-hyperbolic equation, initial-boundary-value problem, spectral method, uniqueness, series, small denominators
Mots-clés : existence, uniform convergence. 
@article{INTO_2017_137_a2,
     author = {K. B. Sabitov and S. N. Sidorov},
     title = {Initial-boundary-value problem for inhomogeneous degenerate equations of mixed parabolic-hyperbolic type},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {26--60},
     publisher = {mathdoc},
     volume = {137},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2017_137_a2/}
}
TY  - JOUR
AU  - K. B. Sabitov
AU  - S. N. Sidorov
TI  - Initial-boundary-value problem for inhomogeneous degenerate equations of mixed parabolic-hyperbolic type
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2017
SP  - 26
EP  - 60
VL  - 137
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/INTO_2017_137_a2/
LA  - ru
ID  - INTO_2017_137_a2
ER  - 
%0 Journal Article
%A K. B. Sabitov
%A S. N. Sidorov
%T Initial-boundary-value problem for inhomogeneous degenerate equations of mixed parabolic-hyperbolic type
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2017
%P 26-60
%V 137
%I mathdoc
%U http://geodesic.mathdoc.fr/item/INTO_2017_137_a2/
%G ru
%F INTO_2017_137_a2
K. B. Sabitov; S. N. Sidorov. Initial-boundary-value problem for inhomogeneous degenerate equations of mixed parabolic-hyperbolic type. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential equations. Mathematical physics, Tome 137 (2017), pp. 26-60. http://geodesic.mathdoc.fr/item/INTO_2017_137_a2/