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Perturbation of semi-linear evolution equations under weak assumptions at initial time. (English) Zbl 1088.34054
This paper builds some perturbation results for initial value problems of the abstract semilinear evolution equation
$\dot{u}(t)+A(t)\,u(t)=f(t,\,u(t)),\qquad t\in (s,\,T],\quad u(s)=u_0,\qquad\qquad$
associated with the perturbated equation of the form
$\dot{u}(t)+A_n(t)\,u(t)=f_n(t,\,u(t)),\qquad t\in (s,\,T],\quad u(s)=u_{0n},$
in a Banach space $$E$$, where $$A(t),\,t\in [0,\,T]$$ and $$A_n(t),\,t\in [0,\,T]$$, are two unbounded closed operator families in $$E$$ generating two evolution systems $$U(t,\,s),\,0\leq s\leq t\leq T$$ and $$U_n(t,\,s),\,0\leq s\leq t\leq T$$. Under rather general conditions, the author shows that the equation and its perturbated equation have unique solutions denoted by $$u$$ and $$u_n$$, respectively. Then, the author establishes the perturbation theorems under very weak assumptions at initial time, for instance, he does not assume that $$u_{0n}\to u_0$$ in $$E$$, not even weakly. As applications of the perturbation theorems, the author proves weak continuity properties and deals with rather general domain perturbation problems for semilinear parabolic and hyperbolic boundary value problems.

##### MSC:
 34G20 Nonlinear differential equations in abstract spaces 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35L20 Initial-boundary value problems for second-order hyperbolic equations
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