On Second-Order Differential Equations with Nonsmooth Second Member

In an abstract framework, we consider the following initial value problem: u′′ + μAu + F(u)u = f  in  (0,T), u(0)=u0,u′(0)=u1, where μ is a positive function and f a nonsmooth function. Given u0, u1, and f we determine Fu in order to have a solution u of the previous equation. We analyze two cases o...

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Bibliographic Details
Published in:ISRN applied mathematics 2014-01, Vol.2014, p.1-13
Main Authors: Miranda, M. Milla, Lourêdo, A. T., Medeiros, L. A.
Format: Article
Language:English
Subjects:
Differential equations
Functions (mathematics)
Hilbert space
Hypotheses
Initial value problems
Linear operators
Mathematical analysis
Theorems
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Summary:In an abstract framework, we consider the following initial value problem: u′′ + μAu + F(u)u = f  in  (0,T), u(0)=u0,u′(0)=u1, where μ is a positive function and f a nonsmooth function. Given u0, u1, and f we determine Fu in order to have a solution u of the previous equation. We analyze two cases of Fu. In our approach, we use the Theory of Linear Operators in Hilbert Spaces, the compactness Aubin-Lions Theorem, and an argument of Fixed Point. One of our two results provides an answer in a certain sense to an open question formulated by Lions in (1981, Page 284).
ISSN:2090-5572
2090-5564
2090-5572
DOI:10.1155/2014/305718