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), , where is a positive function and f a nonsmooth function. Given u0, u1, and f we determine in order to have a solution u of the previous equation. We analyze two cases of . 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). 1. Introduction Let and be two real separable Hilbert spaces with dense in and continuously embedding in . The scalar product and norms of and are represented, respectively, by Let be the self-adjoint operator of defined by the triplet . Consider , . We denote by the Hilbert space equipped with the scalar product (cf. Lions [1]). Consider the following initial value problem: where is a positive function and a nonsmooth function. The objective of this work is to study the following inverse problem: given , , and ( dual space of ) to determine such that Problem (4) has a solution . We analyze two cases of , more precisely, the cases where is an appropriate Hilbert space. In Lions [2, Page 284], the following problem is formulated where is an open bounded set of with boundary , , and being the Dirac mass supported at . He says not to know if this problem admits a solution. He say also that one of the difficulties in the study of existence of solutions of the nonlinear equations lies in the difficulty in defining weak solutions, since the transposition method is essentially a linear method. This is ultimately connected to the fact that one cannot multiply distributions. Problem (4) with of the form (6) is an abstract formulation of Problem (7) with a slight modification of the nonlinear term. Theorem 3 gives the existence of solutions of this problem. In applications we give examples of Problem (7), with the modification of the nonlinear term, for an open bounded set of ,？？ . In Grotta Ragazzo [3] the following equation is studied: This equation is considered as a first approximation of the Klein-Gordon equation Observe that (8) with and is the meson equation of Schiff [4] (cf. also J？rgens [5]). The physical motivation of (8) with can be seen in Lourêdo et al. [6]. Problem (4) with of the form (5) generalizes (8) when . The existence of solutions of this problem is studied in Theorem 1. In Louredo et al., loc.cit., is analyzed the equation with nonlinear boundary condition. The given in (5) is different from the of