Bounded solutions of parabolic equations in continuous function spaces. (English) Zbl 1151.34049

Liu, James; N’Guérékata, Gaston; Nguyen Van Minh; Vu Quoc Phong

Funkc. Ekvacioj, Ser. Int. 49, No. 3, 337-355 (2006).

Summary: This paper is concerned with the existence of bounded mild solutions to equations of the form
\[ u^{\prime}(t) = Au(t) + f(t), \] where \(A\) generates a holomorphic semigroup that is not necessarily strongly continuous, and \(f\) is a bounded function. This problem arises when one considers a parabolic equation in spaces of continuous functions. The obtained results, that are stated in terms of spectral properties of the spectrum of \(A\) and the uniform spectrum of \(f\), extend previous ones.

MSC:

34G10	Linear differential equations in abstract spaces
35K90	Abstract parabolic equations

Keywords:

parabolic equation; continuous function space; complete second order evolution equation; mild solution