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Identification for a semilinear evolution equation in a Banach space. (English) Zbl 1205.34072

The following identification problem for a first-order semilinear differential equation in a general Banach space \(X\) is considered
\[ u'(t)=Au(t)+f(t)+\varphi(u(t))z,\quad t\in(0,1),\;u(0)=\xi_0,\;\int^1_0 u(t)dt=\xi_1, \]
where \(A\) is the infinitesimal generator of a \(C_0\)-semigroup of contractions, \(f\) is a \(C^1\) function, \(\varphi\) is a \(C^1\) functional, \(\xi_0, \xi_1\in X\), and \(z\in X\) and \(u\) are unknown. The authors prove the existence of a strict solution to the considered problem admitting an implicit representation. Moreover, under the additional assumption that the \(C_0\)-semigroup generated by the linear part has a sufficiently fast exponential decay, the uniqueness and the continuous dependence on the data of the solution are established. As an application, a semilinear parabolic problem is studied.

MSC:

34G20 Nonlinear differential equations in abstract spaces
47D06 One-parameter semigroups and linear evolution equations
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