2022/01/13 by Aras Bacho, Bacho, Aras
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Differential Equations Analysis #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2201.05235
openalex publication_date 2022/01/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The well-posedness of the abstract Cauchy problem for the doubly nonlinear evolution inclusion equation of second order \begincases u''(t)+∂ Ψ(u'(t))+B(t,u(t))\ni f(t), amp; t∈ (0,T), Tgt;0,
u(0)=u0, u'(0)=v0 \endcases in a real separable Hilbert space \mathscrH, where u0∈ \mathscrH, v0∈ D(∂ Ψ)∩ D(Ψ), f∈ L2(0,T;\mathscrH). The functional Ψ: \mathscrH → (-∞,+∞] is supposed to be proper, lower semicontinuous, and convex and the nonlinear operator B:[0,T]× \mathscrH→ \mathscrH is supposed to satisfy a (local) Lipschitz condition. Existence and uniqueness of strong solutions u∈ H2(0,T^*;\mathscrH) as well as the continuous dependence of solutions from the data re shown by employing the theory of nonlinear semigroups and the Banach fixed-point theorem. If B satisfies a local Lipschitz condition, then the existence of strong local solutions are obtained.