Article

Full entry | PDF   (1.8 MB)
Keywords:
partial differential equations
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.
References:
[1] P. Fife: Solutions of parabolic boundary problems existing for all time. Arch. Rat. Mech. Anal. 76, 1964, 155-186. MR 0167727 | Zbl 0173.38204
[2] И. И. Шмулев: Периодические решения первой краевой задачи для параболических уравнений. Математический сборник 66 (108), 3, 1965, 398-410. MR 0173097 | Zbl 1099.01519
[3] J. L. Lions: Sur certain équations paraboliques non linéaires. Bull. Soc. Math. France, 93, 2, 1965, 155-176. MR 0194760
[4] T. Kusano: A remark on a periodic boundary problem of parabolic type. Proc. Jap. Acad. XLII, I, 1966, 10-12. MR 0211034 | Zbl 0166.37102
[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. MR 0209684 | Zbl 0154.36101
[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

Partner of