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On the mildly degenerate Kirchhoff string
Author(s) -
Arosio A.,
Garavaldi S.
Publication year - 1991
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1670140303
Subject(s) - mathematics , lipschitz continuity , degenerate energy levels , initial value problem , cauchy problem , mathematical analysis , string (physics) , uniqueness , hilbert space , pure mathematics , order (exchange) , space (punctuation) , mathematical physics , combinatorics , physics , quantum mechanics , linguistics , philosophy , finance , economics
Abstract Let us consider the Cauchy problem for the abstract evolution equation, which describes the Kirchhoff sring u ″ + m (〈 Au, u 〉) Au = 0, ( t > 0),Where ( )′ = d/dt, A is any symmetric isomorphism of a Hilbert space V into its (anti) dual V ′ and m is a function of one real variable. Following physical considerations, we allow m (·) to be any non‐decreasing, non‐negative continuous function (a degenerate Kirchhoff string). We assume that the initial value u 0 satisfies: m (〈 Au 0 , u 0 〉)>0 (a mildly degenerate Kirchhoff string), that m is locally Lipschitz continuous in the open region where it does not vanish, and that m has a finite order of vanishing (e.g., m (ρ) = ρ γ , γ > 0). We establish the local well‐posedness of the Cauchy problem in the class D ( A α/2 ) × D( A (α−1)/2 ), For each α ⩾ 3/2. This improves the results of Medeiros and Miranda [15], Ebihara et al .[9] and a result of Yamada [27]. We are not able to prove the global existence of the solution; however, we provide a lower estimate for the life span of the solution, which yields the almost global existence (AGE) in the case when m (ρ) = ρ γ (γ>0).