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On second-order differential equations with nonsmooth second member. (English) Zbl 1304.35444
Summary: In an abstract framework, we consider the following initial value problem: \(u''+\mu Au+F(u)u=f\) in \((0,T)\), \(u(0)=u^0\), \(u'(0)=u^1\), where \(\mu\) is a positive function and \(f\) a nonsmooth function. Given \(u^0\), \(u^1\), and \(f\) we determine \(F(u)\) in order to have a solution \(u\) of the previous equation. We analyze two cases of \(F(u)\). In our approach, we use the theory of linear operators in Hilbert spaces, the compactness Aubin-Lions theorem, and an fixed point argument. One of our to results provides an answer in a certain sense to an open question formulated by J. L. Lions on p. 284 of [Some methods in the mathematical analysis of systems and their control. Beijing, China: Science Press; New York: Gordon and Breach, Science Publishers (1981; Zbl 0542.93034)].

MSC:
35L90 Abstract hyperbolic equations
35L71 Second-order semilinear hyperbolic equations
35L20 Initial-boundary value problems for second-order hyperbolic equations
34G20 Nonlinear differential equations in abstract spaces
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