Geodesic
Boundary behavior of solutions to the Dirichlet problem for the heat equation in a domain whose lateral boundary satisfies the H\"older condition with exponent less than $1/2$
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Algebra, geometry, differential equations, Tome 217 (2022), pp. 37-40.

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

For the heat equation with one space variable, we examine solutions of the first boundary-value problem in a domain whose lateral boundary possesses a model singularity, namely, the curve describing the lateral boundary is smooth everywhere except for one point and belongs to the Hölder class with exponent less than $1 /2$. We prove that if a solution is positive in some neighborhood of the singular point and vanishes on the lateral boundary in this neighborhood, then the first derivative of this solution unboundedly increases in any neighbourhood of the singular point.
Keywords: heat equation, first boundary-value problem, nonsmooth lateral boundary, barrier method. 
@article{INTO_2022_217_a4,
     author = {A. N. Konenkov},
     title = {Boundary behavior of solutions to the {Dirichlet} problem for the heat equation in a domain whose lateral boundary satisfies the {H\"older} condition with exponent less than $1/2$},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {37--40},
     publisher = {mathdoc},
     volume = {217},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2022_217_a4/}
}
TY  - JOUR
AU  - A. N. Konenkov
TI  - Boundary behavior of solutions to the Dirichlet problem for the heat equation in a domain whose lateral boundary satisfies the H\"older condition with exponent less than $1/2$
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2022
SP  - 37
EP  - 40
VL  - 217
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/INTO_2022_217_a4/
LA  - ru
ID  - INTO_2022_217_a4
ER  - 
%0 Journal Article
%A A. N. Konenkov
%T Boundary behavior of solutions to the Dirichlet problem for the heat equation in a domain whose lateral boundary satisfies the H\"older condition with exponent less than $1/2$
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2022
%P 37-40
%V 217
%I mathdoc
%U http://geodesic.mathdoc.fr/item/INTO_2022_217_a4/
%G ru
%F INTO_2022_217_a4
A. N. Konenkov. Boundary behavior of solutions to the Dirichlet problem for the heat equation in a domain whose lateral boundary satisfies the H\"older condition with exponent less than $1/2$. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Algebra, geometry, differential equations, Tome 217 (2022), pp. 37-40. http://geodesic.mathdoc.fr/item/INTO_2022_217_a4/

[1] Arefev V. N., Bagirov L. A., “O resheniyakh uravneniya teploprovodnosti v oblastyakh s osobennostyami”, Mat. zametki., 64:2 (1998), 163–179 | MR

[2] Kozlov V. A., Mazya V. G., “Ob osobennostyakh resheniya pervoi kraevoi zadachi dlya uravneniya teploprovodnosti v oblastyakh s konicheskoi tochkoi”, Izv. vuzov. Mat., 1987, no. 2, 38–46

[3] Petrovskii I. G., “O reshenii pervoi kraevoi zadachi dlya uravneniya teploprovodnosti”, Uch. zap. MGU., 1934, no. 2, 55–59