Geodesic
Stabilization of solutions of the third mixed problem for a second order parabolic equation in a noncylindrical domain
Sbornik. Mathematics, Tome 39 (1981) no. 1, pp. 87-105
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

This paper studies the behavior for large values of time $t$ of the solution of the third mixed problem in a noncylindrical domain $D\subset\mathbf R^{n+1}$ that expands as $t$ increases, for a linear second order parabolic equation in selfadjoint form without lower terms. In this connection the boundary condition is chosen so that the “energy conservation law” holds. For a very large class of domains a simple geometric characteristic of the domain is singled out-the function $V(t,\sqrt t)=\operatorname{mes}_n(D_t\cap\{|x|<\sqrt t\})$, where $D_t$ is the intersection of the domain $D$ with the hyperplane $t=\operatorname{const}$ – determining the stabilization speed of the solution. Namely, it is proved that a solution $u(t,x)$ of the above problem with initial function $\varphi$ from $L_1(D_0)$ satisfies the estimate $$ \|u(t,x)\|_{L_\infty(D_t)}\leqslant\frac C{V(t,\sqrt t)}\|\varphi\|_{L_1(D_0)},\qquad t>0, $$ and the accuracy of this estimate is of the order of the convergence to zero as $t\to\infty$. Bibliography: 6 titles. 
@article{SM_1981_39_1_a3,
     author = {V. I. Ushakov},
     title = {Stabilization of solutions of the third mixed problem for a~second order parabolic equation in a~noncylindrical domain},
     journal = {Sbornik. Mathematics},
     pages = {87--105},
     year = {1981},
     volume = {39},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1981_39_1_a3/}
}
TY  - JOUR
AU  - V. I. Ushakov
TI  - Stabilization of solutions of the third mixed problem for a second order parabolic equation in a noncylindrical domain
JO  - Sbornik. Mathematics
PY  - 1981
SP  - 87
EP  - 105
VL  - 39
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/SM_1981_39_1_a3/
LA  - en
ID  - SM_1981_39_1_a3
ER  - 
%0 Journal Article
%A V. I. Ushakov
%T Stabilization of solutions of the third mixed problem for a second order parabolic equation in a noncylindrical domain
%J Sbornik. Mathematics
%D 1981
%P 87-105
%V 39
%N 1
%U http://geodesic.mathdoc.fr/item/SM_1981_39_1_a3/
%G en
%F SM_1981_39_1_a3
V. I. Ushakov. Stabilization of solutions of the third mixed problem for a second order parabolic equation in a noncylindrical domain. Sbornik. Mathematics, Tome 39 (1981) no. 1, pp. 87-105. http://geodesic.mathdoc.fr/item/SM_1981_39_1_a3/

[1] A. K. Guschin, “Ob otsenkakh resheniya kraevykh zadach dlya parabolicheskogo uravneniya vtorogo poryadka”, Trudy Matem. in-ta im. V. A. Steklova, XXVI (1973), 5–45

[2] A. K. Guschin, “Stabilizatsiya reshenii vtoroi kraevoi zadachi dlya parabolicheskogo uravneniya vtorogo poryadka”, Matem. sb., 101(143) (1978), 459–499

[3] V. I. Ushakov, “O povedenii reshenii tretei smeshannoi zadachi dlya parabolicheskikh uravnenii vtorogo poryadka pri $t\to\infty$”, Diff. uravneniya, 15:2 (1979), 310–320 | MR | Zbl

[4] A. K. Guschin, “Nekotorye svoistva obobschennogo resheniya vtoroi kraevoi zadachi dlya parabolicheskogo uravneniya”, Matem. sb., 97(139) (1977), 242–261

[5] A. K. Guschin, “Ob otsenkakh integrala Dirikhle v neogranichennykh oblastyakh”, Matem. sb., 99(141) (1977), 282–294

[6] E. De Giorgi, “Sulla differenziabilità e I'analiticità delle estremali degli integrali multipli regolari”, Mem. Accad. Sci. Torino, ser. 3, 3 (1957), 25–43 ; Математика, 4:6 (1960), 23–28 (русский перевод) | MR | Zbl