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On the asymptotic behavior of the solutions of semilinear nonautonomous equations. (English) Zbl 1280.34055

The paper is on semilinear equations \[ x'(t) = A(t)x(t) + f(t, x(t)) \quad (t \geq s) \, , \qquad x(s) = x, \tag{1} \] with \(\{A(t)\}\) a family of (generally unbounded) operators in a Banach space \(X\). The linear equation \((f = 0)\) is well posed, which means the existence of a family \(\{U(t, s)\}\) of linear bounded operators such that for every \(s\) the function \(x(t) = U(t, s)x\) is the unique solution of (1). The family \(\{U(t, s)\}\) is called the evolution operator of the linear equation and is assumed strongly continuous in \(s, t;\) in infinite dimension, “solution” is usually understood in a weak sense. If \(A(t) = A\) then \(U(t, s) = S(t - s)\), \(S(t)\) the strongly continuous semigroup generated by \(A\).
Solutions of the full equation (1) are by definition solutions of the integral equation coming from the variation-of-constants formula, \[ x(t) = U(t, s)x + \int_s^t U(t, \tau)f(\tau, x(\tau))d\tau \;\quad (t \geq s), \tag{2} \] where \(f(t, x)\) is assumed jointly continuous in \(t, x\). In general, a Lipschitz condition on \(f(t, x)\) with respect to \(x\) guarantees local existence and uniqueness; here the Lipschitz condition is uniform, thus the solutions \(x(t)\) of (2) are global. The evolution operator of the full equation (1) is defined from the solutions by \(X(t, s)x = x(t)\), and the subject is exponential decay of \(X(t, s)\): this means \[ \|X(t, s)\|_{\text{lip}} \leq Ce^{-\omega(t - s)} \tag{3} \] with \(C, \omega > 0\), where \(\|T\|_{lip}\) is the infimum of all \(M \geq 0\) such that \(\|Tx - Ty\| \leq M\|x - y\|\) for all \(x, y \in X\). The authors give a condition on Green’s operator \[ {\mathcal G}(t, s) = \int_0^t X(t, s)f(s)ds \] which guarantees (3); roughly it requires \(\mathcal G\) to map \(L^p(0, \infty; X)\) into \(L^q(0, \infty; X)\) Lipschitz continuously, where \(p, q \in [1, \infty]\), \(\{p, q\} \neq \{1, \infty\}\).
The ancestry of this paper lies in classical 1930 results by Perron (for ordinary differential equations) generalized by Massera and Schäffer and Daleckii and Krein to infinite-dimensional equations.

MSC:

34D05 Asymptotic properties of solutions to ordinary differential equations
34G20 Nonlinear differential equations in abstract spaces
34D20 Stability of solutions to ordinary differential equations
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References:

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