# zbMATH — the first resource for mathematics

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
##### Keywords:
compactness Aubin-Lions theorem
Full Text:
##### References:
 [1] Lions, J.-L., Équations aux Dérivées Partielles-Interpolation. Vol. I, (2003), EDP Sciences, Les Ulis, Paris, France [2] Lions, J.-L., Some Methods in the Mathematical Analysis of System and Their Control, (1981), Science Press, Beijing, China; Gordon Breach, Science Publishers, New York, NY, USA · Zbl 0542.93034 [3] Grotta Ragazzo, C., Chaos and integrability in a nonlinear wave equation, Journal of Dynamics and Differential Equations, 6, 1, 227-244, (1994) · Zbl 0802.34044 [4] Schiff, L. I., Nonlinear meson theory of nuclear forces. I. neutral scalar mesons with point-contact repulsion, The Physical Reviews, 84, 1, 1-9, (1951) · Zbl 0054.08305 [5] Jörgens, K., Des aufangswert problem in grossen für eine klasse nichtlinearer wellengleichungen, Mathematische Zeitschrift, 77, 295-308, (1971) [6] Lourêdo, A. T.; Araújo, M. A. F.; Milla Miranda, M., On a nonlinear wave equation with boundary damping, Mathematical Methods in the Applied Sciences, (2013) · Zbl 1292.35049 [7] Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations, 44, (1993), Springer, New York, NY, USA · Zbl 0516.47023 [8] Lions, J. L.; Magenes, E., Problèmes Aux Limites Non-Homogènes et Applications. Vol. I, (1968), Dunod, Paris, France · Zbl 0165.10801 [9] Brezis, H., Functional Analysis, Sobolev Spaces and Partial Differential Equations, (2010), Springer, New York, NY, USA [10] Komornik, V., Exact Controllability and Stabilization. The Multiplier Method, (1994), John Wiley and Sons, Paris, France · Zbl 0937.93003 [11] Medeiros, L. A.; Milla Miranda, M., Espaços de Sobolev (Iniciação aos Problemas Elíticos Não-Homogêneos), (2006), IM-UFRJ, Rio de Janeiro, Brazil [12] Lions, J. L., Quelques Méthodes De Résolution des Problémes aux Limites Non-Linéaires, (1969), Dunod, Paris, France · Zbl 0189.40603 [13] Vicente, A.; Frota, C. L., On a mixed problem with a nonlinear acoustic boundary condition for a non-locally reacting boundaries, Journal of Mathematical Analysis and Applications, 407, 2, 328-338, (2013) · Zbl 1310.35153 [14] Simon, J., Compact sets in the space $$L^p(0, T; B)$$, Annali di Matematica Pura ed Applicata, 146, 65-96, (1987) · Zbl 0629.46031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.