Periodic solutions of the first boundary value problem for a linear and weakly nonlinear heat equation

Věnceslava Šťastnová; Otto Vejvoda

Aplikace matematiky (1968)

  • Volume: 13, Issue: 6, page 466-477
  • ISSN: 0862-7940

Abstract

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One investigates the existence of an ω -periodic solution of the problem u t = u x x + c u + g ( t , x ) + ϵ f ( t , x , u , u x , ϵ ) , u ( t , 0 ) = h 0 ( t ) + ϵ χ 0 ( t , u ( t , 0 ) , u ( t , π ) ) , u ( t , π ) = h 1 ( t ) + ϵ χ 1 ( t , u ( t , 0 ) , u ( t , π ) ) , provided the functions g , f , h 0 , h 1 , χ 0 , χ 1 are sufficiently smooth and ω -periodic in t . If c k 2 , k natural, such a solution always exists for sufficiently small ϵ > 0 . On the other hand, if c = l 2 , l natural, some additional conditions have to be satisfied.

How to cite

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Šťastnová, Věnceslava, and Vejvoda, Otto. "Periodic solutions of the first boundary value problem for a linear and weakly nonlinear heat equation." Aplikace matematiky 13.6 (1968): 466-477. <http://eudml.org/doc/14568>.

@article{Šťastnová1968,
abstract = {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 $c\ne k^2$, $k$ 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.},
author = {Šťastnová, Věnceslava, Vejvoda, Otto},
journal = {Aplikace matematiky},
keywords = {partial differential equations},
language = {eng},
number = {6},
pages = {466-477},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Periodic solutions of the first boundary value problem for a linear and weakly nonlinear heat equation},
url = {http://eudml.org/doc/14568},
volume = {13},
year = {1968},
}

TY - JOUR
AU - Šťastnová, Věnceslava
AU - Vejvoda, Otto
TI - Periodic solutions of the first boundary value problem for a linear and weakly nonlinear heat equation
JO - Aplikace matematiky
PY - 1968
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 13
IS - 6
SP - 466
EP - 477
AB - 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 $c\ne k^2$, $k$ 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.
LA - eng
KW - partial differential equations
UR - http://eudml.org/doc/14568
ER -

References

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  1. P. Fife, Solutions of parabolic boundary problems existing for all time, Arch. Rat. Mech. Anal. 76, 1964, 155-186. (1964) Zbl0173.38204MR0167727
  2. И. И. Шмулев, Периодические решения первой краевой задачи для параболических уравнений, Математический сборник 66 (108), 3, 1965, 398-410. (1965) Zbl1099.01519MR0173097
  3. J. L. Lions, Sur certain équations paraboliques non linéaires, Bull. Soc. Math. France, 93, 2, 1965, 155-176. (1965) MR0194760
  4. T. Kusano, A remark on a periodic boundary problem of parabolic type, Proc. Jap. Acad. XLII, I, 1966, 10-12. (1966) Zbl0166.37102MR0211034
  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. (1966) Zbl0154.36101MR0209684
  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. (1964) Zbl0178.45302MR0174872

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