Stability in abstract functional differential equations. II: Applications (Q1340527)

In part I [J. Math. Anal. Appl. 186, No. 2, 534-558 (1994; Zbl 0814.34066)], theorems of stability in the sense of \textit{J. Kato} [Stability in functional differential equations, Functional differential equations and bifurcation, Proc. Conf., Sao Carlos/Brazil 1979, Lect. Notes Math. 799, 252-262 (1980; Zbl 0437.34059)] were demonstrated. The second part of the paper is devoted to some models on both ordinary and partial differential equations with bounded delays. Stability of the following scalar equations is investigated: 1. \(u_ t'= u_{xx}(t, x)+ wu(t, x)+ f(u(t- h, x)), u(t, 0)= u(t, \pi)= 0\), \(f(0)= 0\), for \(t\geq 0\), \(0\leq x\leq \pi\), 2. \(u_{tt}'= (u_ x+ u^ 3_ x)_ x- \alpha u_ t'+ C(t) u(t- h, x)\), \(u(t, 0)= u(t, 1)= 0\), 3. \(u_ t'= a(t) u_{xx}+ b(t) u_{xx}(t- h, x)\), \(u(t, 0)= u(t, \pi)\), \(t\geq 0\), \(x\in [0, \pi]\), 4. \(x'(t)= - a(t) x(t)+ b(t)\int^ t_{t- h} x(u) du\), 5. \(x'(t)= - a(t) x(t)+ \int^ t_{t- h} b(s) x(s)ds\).