Solution Estimates for Semilinear Non-autonomous Evolution Equations with Differentiable Linear Parts
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Solution Estimates for Semilinear Non-autonomous Evolution Equations with Differentiable Linear Parts Michael Gil’1 © Foundation for Scientific Research and Technological Innovation 2019
Abstract In a Hilbert space we consider the equation du(t)/dt = A(t)u(t) + F(t, u(t)), where A(t) is an unbounded operator, having a bounded strong derivative, and F is a continuous mapping. We derive norm estimates for solutions of the considered equation. These estimates give us the L 2 -stability and absolute stability conditions. To the best of our knowledge an absolute stability test for nonautonomous evolution equations has been obtained for the first time. Our main tool is the norm estimate for the derivative of a solution of the time-dependent Lyapunov equation. Keywords Semi-linear differential equation · Non-autonomous equation · Hilbert space · Absolute stability · L 2 -stability · Quasi-linear equation Mathematics Subject Classification 34G20 · 37L15 · 35K58 · 35K59
Introduction and Statement of the Main Result We deal with the stability of solutions to semilinear differential equations in a Hilbert space. The literature on stability of semilinear differential equations in Banach and Hilbert spaces is very rich, but mainly equations with the autonomous linear parts have been investigated, cf. [1,3,4,9,12,14,15] and references given therein. In the paper [11] a linear differential equation with an unbounded operator A(t) in a Hilbert space having a bounded strong derivative has been considered. For that equation solution estimates and stability conditions have been obtained. The aim of the present paper is to generalize the main result from [11] to a class of of semilinear non-autonomous differential equations in a Hilbert space. For the considered equations we derive norm estimates of solutions. These estimates give us the L 2 -stability and absolute stability conditions. To the best of our knowledge, the absolute stability conditions for abstract semilinear nonautonomous differential equations have not been investigated in the available literature, cf.
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Michael Gil’ [email protected] Department of Mathematics, Ben Gurion University of the Negev, P.O. Box 653, Be’er-Sheva 84105, Israel
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Differential Equations and Dynamical Systems
[12,14]. In addition, the L 2 -stability conditions in the terms of the derivative of the linear part obtained below are new. √ Let H be a complex Hilbert space with a scalar product ., ., the norm . = ., . and unit operator I . For a linear operator C, C ∗ is the adjoint one, Dom(C) is the domain. L(H) denotes the set of all bounded operators in H with the norm · L(H) . Throughout this paper A(t) (t ≥ 0) is a closed operator on H with a constant dense domain D0 having a strong derivative A (t) uniformly bounded on [0, ∞); L 2 ([0, ∞); H) is the space of functions w : [0, ∞) → H with the finite norm ∞ 1/2 2 w L 2 = w(t) dt , 0
C([0, ∞); H) is the space of continuous functions w : [0, ∞) → H with the finite norm wC(0,∞) = supt≥0 w(t). In addition, (r ) :=
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