Gheorghe Moroșanu†, Adrian Petrușel‡
Abstract: In a real Hilbert space H, let us consider the boundary-value problem −εu 00 (t) + µu 0 (t) + Au(t) + Bu(t) 3 f(t), t ∈ [0, T]; u(0) = u0, u 0 (T) = 0, where T > 0 is a given time instant, ε, µ are positive parameters, A : D(A) ⊂ H → H is a (possibly set-valued) maximal monotone operator, and B : H → H is a Lipschitz operator. In this paper, we investigate the behavior of the solutions to this problem in two cases: (i) µ > 0 fixed, 0 < ε → 0, and (ii) ε > 0 fixed and 0 < µ → 0. Notice that if µ = 1 and ε is a positive small parameter, the above problem is a Lions-type regularization of the Cauchy problem u 0 (t) + Au(t) + Bu(t) 3 f(t), t ∈ [0, T]; u(0) = u0, which was recently studied by L. Barbu and G. Moro¸sanu [Commun. Contemp. Math. 19 (2017)]. Our abstract results are illustrated with examples related to the heat equation and the telegraph differential system.
MSC: 34G25, 47J35, 47H05, 35K20, 35L50
keywords: Lions regularization, approximation, maximal monotone operator, Lipschitz operator, heat equation, telegraph differential system.
DOI 10.56082/annalsarscimath.2020.1-2.274
† morosanu@math.ubbcluj.ro, Faculty of Mathematics and Computer Science, Babeș Bolyai University, Cluj-Napoca, Romania & Academy of Romanian Scientists
‡petrusel@math.ubbcluj.ro, Faculty of Mathematics and Computer Science, Babeș Bolyai University, Cluj-Napoca, Romania & Academy of Romanian Scientists
PUBLISHED in Annals Academy of Romanian Scientists Series on Mathematics and Its Application, Volume 12 no 1-2, 2020
