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