On the mildly degenerate Kirchhoff string

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).