On the mildly degenerate Kirchhoff string

✍ Scribed by A. Arosio; S. Garavaldi

• Book ID: 102948245
• Publisher: John Wiley and Sons
• Year: 1991
• Tongue: English
• Weight: 675 KB
• Volume: 14
• Category: Article
• ISSN: 0170-4214
• DOI: 10.1002/mma.1670140303

✦ Synopsis

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).