| Title:
|
Periodic solutions of the first boundary value problem for a linear and weakly nonlinear heat equation (English) |
| Author:
|
Šťastnová, Věnceslava |
| Author:
|
Vejvoda, Otto |
| Language:
|
English |
| Journal:
|
Aplikace matematiky |
| ISSN:
|
0373-6725 |
| Volume:
|
13 |
| Issue:
|
6 |
| Year:
|
1968 |
| Pages:
|
466-477 |
| Summary lang:
|
English |
| Summary lang:
|
Czech |
| . |
| Category:
|
math |
| . |
| Summary:
|
One investigates the existence of an $\omega$-periodic solution of the problem $u_t=u_{xx}+cu+g(t,x)+\epsilon f(t,x,u,u_x,\epsilon),\ u(t,0)=h_0(t)+\epsilon \chi_0(t,u(t,0),u(t,\pi)), u(t,\pi)=h_1(t)+\epsilon \chi_1(t,u(t,0), u(t,\pi))$, provided the functions $g,f,h_0,h_1,\chi_0,\chi_1$ are sufficiently smooth and $\omega$-periodic in $t$. If $c\neq k^2$, $k$ natural, such a solution always exists for sufficiently small $\epsilon >0$. On the other hand, if $c=l^2$, $l$ natural, some additional conditions have to be satisfied. (English) |
| Keyword:
|
partial differential equations |
| MSC:
|
35-12 |
| idZBL:
|
Zbl 0165.44302 |
| idMR:
|
MR0243188 |
| DOI:
|
10.21136/AM.1968.103196 |
| . |
| Date available:
|
2008-05-20T17:43:49Z |
| Last updated:
|
2020-07-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/103196 |
| . |
| Reference:
|
[1] P. Fife: Solutions of parabolic boundary problems existing for all time.Arch. Rat. Mech. Anal. 76, 1964, 155-186. Zbl 0173.38204, MR 0167727, 10.1007/BF00250642 |
| Reference:
|
[2] И. И. Шмулев: Периодические решения первой краевой задачи для параболических уравнений.Математический сборник 66 (108), 3, 1965, 398-410. Zbl 1099.01519, MR 0173097 |
| Reference:
|
[3] J. L. Lions: Sur certain équations paraboliques non linéaires.Bull. Soc. Math. France, 93, 2, 1965, 155-176. MR 0194760 |
| Reference:
|
[4] T. Kusano: A remark on a periodic boundary problem of parabolic type.Proc. Jap. Acad. XLII, I, 1966, 10-12. Zbl 0166.37102, MR 0211034 |
| Reference:
|
[5] T. Kusano: Periodic solutions of the first boundary problem for quasilinear parabolic equations of second order.Funkc. Ekvac. 9, 1 - 3, 1966, 129-138. Zbl 0154.36101, MR 0209684 |
| Reference:
|
[6] O. Vejvoda: Periodic solutions of a linear and weakly nonlinear wave equation in one dimension.I. Czech. Math. J. 14 (89), 1964, 341-382. MR 0174872 |
| . |
