A class of non-analytic functions for the global solvability of Kirchhoff equation

Abstract

We consider the global solvability to the Cauchy problem of Kirchhoff equation with generalized classes of Manfrin's class. Manfrin's class is a subclass of Sobolev space, but we shall extend this class as a subclass of the ultradifferentiable class, and we succeed to prove the global solvability of Kirchhoff equation with large data in wider classes from the previous works. MSC 2000: 35L70, 35L15..• Noting lim r→∞ q(r; ρ) = 1, p(|ξ|; ρ)q(|ξ|; ρ) is the usual weight of Sobolev type outside of Ω(ρ).• q(|ξ|; ρ) has influence only near to |ξ| = ρ, and the effective area is smaller as m larger. However, the influence of q(|ξ|; ρ) cannot be ignored since ρ → ∞.For strictly increasing positive continuous functions M and M, we consider a generalization of G m (ρ, η) as follows:Indeed, we see that G(ρ, η, M, M) = G m (ρ, η) for M(r) = M(r) = r m . The main purpose of the present paper is to extend the order of M(r), M(r) from polynomial to infinite order.

A road map to the blow-up for a Kirchhoff equation with external force

“…The same is true whenever the eigenvalues of the operator A are a sequence that grows fast enough. We refer to [13,14,18,19,21] for precise statements. For the sake of completeness, we point out that the spectral gap theory has been recently extended in order to show the existence of global weak solutions in the energy space D(A 1/2 ) × H (see [10]).…”

Almost global existence for Kirchhoff equations around global solutions

“…Existence and uniqueness of local solutions behind this regularity threshold, and in particular for initial data in the so-called energy space D(A 1/2 ) × H, is still a largely open problem (the only result we are aware of in this direction is contained in the recent paper [9]). Global-in-time solutions to problem (1.1)- (1.3) are known to exist in many different special cases, involving either special initial data (such as analytic data [5,2,6,7], quasianalytic data [18,11], lacunary data [16,13,10,11,14]), or special nonlinearities [19], or dispersive equations and small data [12,8,20,17], or spectral gap operators [11].…”

Approximate Inertial Manifold for a Class of the Kirchhoff Wave Equations with Nonlinear Strongly Damped Terms

“…Kirchhoff type wave equations have been studied by many scholars (see [9] [10] [11]). In reference [12], the long time behavior of solutions for the initial value problems (1.…”