On the Existence and Growth of Mild Solutions of the Abstract Cauchy Problem for Operators with Polynomially Bounded Resolvent

نویسنده

  • GILES AUCHMUTY
چکیده

In this paper we study the growth of mild solutions of abstract Cauchy problems governed by a densely defined generator A of an o-times integrated semigroup {Sa(t)},>o. We prove the following results: (i) If lIsa( I Me wt for ime M > 0, w E W, and all t 2 0, then for all E > 0, o > 0 and zo E D((-Aw+o)a+E) a unique mild solution exists. Moreover, this solution is exponentially bounded, and its exponential type is at most w. If zo E D((-A,+,)l+a+E), the solution is classical. (ii) If IISa(t)jl 5 M(l+t’) for some constants M 2 1, y 2 0, and all t 2 0, then for all E > 0, o > 0 and all zo E D((-Au)*+‘) a unique mild solution exists. Moreover, this solution is polynomially bounded, and its polynomial type is at most max{cy If E, y + E, 2-7 cy + E}. If zo E D((-Ao)l+“+‘), the solution is classical. These results are applied to study the growth of mild solutions of the Cauchy problem governed by a densely defined operator whose resolvent is polynomially bounded in the open right half plane.

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تاریخ انتشار 2003