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On Hopf bifurcation theorem for parabolic equations with infinite delay. (English) Zbl 0749.35007

The author discusses the existence of the Hopf bifurcation for parabolic functional equations with infinite delay: \[ u(t)=Au(t)+Lu_ t+f(\mu,u(t),u_ t), \] where \(u_ t(s)=u(t+s)\) for \(s\in R^ -\). \(A\) is a generator of an analytic semigroup on a Banach space \(X\), \(L\) is a continuous linear operator from an appropriate function space \(Y\) into \(X\), \(f\in C^ 2([-1,1]\times{\mathcal D}((-A)^ \alpha)\times Y,X)\). He generalizes results of G. Da Prato and A. Lunardi [Indiana Univ. Math. J. 36, 241-255 (1987; Zbl 0634.45013)], A. Tesei [Ann. Mat. Pura Appl., IV Ser. 126, 103-115 (1980; Zbl 0463.45009)] and Y. Yamada and Y. Niikura [Funkc. Ekvacioj, Ser. Int. 29, 309-333 (1986; Zbl 0624.35007)], needing less restrictions on delay terms \(Lu_ t\) and on the smoothness of the data, and applies them to an integro- differential equation with a singular kernel: \[ u_ t=au_{xx}+bu+c\int^ t_{-\infty}k(t-s)u_{xx}(s)ds+\int^ t_{- \infty}k_ 1(t-s)(g (\mu,u(s)_ x)_ x)ds \]
\[ u(t,0)=u(t,\pi)=0,\quad t\in\mathbb{R}, \] where \(k(t)=t^{-\gamma}e^{-pt}\) with \(\gamma<(2- \alpha)/3\), \(p>0\), \(g\) is smooth with \(g_ u(0,0)=0\), \(k_ 1\in L^ 1\) and \(g_{\mu u}(0,0)\hat k_ 1(i)=0\). In the proof he applies the idea of M. G. Crandall and P. H. Rabinowitz [Arch. Ration. Mech. Anal. 52, 161-180 (1973; Zbl 0275.47044)] and decomposes the space of periodic functions before an application of the implicit function theorem.
Reviewer: K.Yoshida

MSC:

35B32 Bifurcations in context of PDEs
35R10 Partial functional-differential equations
45K05 Integro-partial differential equations
35B10 Periodic solutions to PDEs

Keywords:

singular kernel
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