ABSTRACT
We study asymptotic behavior and singularity blow-up formation phenomena in the Cauchy problem for the fourth-order Boussinesq-type, or a semilinear wave (hyperbolic), equation (the SWE-4, for short)
utt = −uxxxx + |u|p−1u in IR × IR+, (1) where p > 1 is a fixed exponent. A local existence-uniqueness of a solution of the Cauchy problem with smooth bounded initial data is a standard question of PDE theory, which was resolved a long time ago using various well-developed approaches. However, in a global time-setting, there is a difficulty: due to a superlinear
growth of the lower-order term |u|p−1u for u 1, the solutions of the Cauchy problem for (1) may blow up in finite time, as t → T− < ∞. Such global nonexistence phenomena for semilinear and quasilinear higher-order wave and hyperbolic equations have been extensively studied in the mathematical literature; see Mitidieri-Pohozaev’s book [303, Part III] for main references and results on conditions of blow-up by nonlinear capacity and other approaches. In general, our intention is to study a possible “micro-scale” blow-up struc-
ture of solutions of the higher-order PDE (1). To this end, we perform a blow-up scaling at a fixed point {x = 0, t = T } to derive the rescaled equation and apply necessary Hermitian spectral theory developed. The most
and
interesting local patterns occur at the blow-up points T < ∞, at which lim
|u(x, t)| = +∞. (2)
In this case, we can observe both linearized and nonlinear blow-up patterns. It is important that if the solution is bounded at t = T , i.e., is regular, then all the existing patterns as multiple zero structures are always linearized and their local structures are given by generalized Hermite polynomials of any finite order, which are eigenfunctions of a quadratic pencil of non-self-adjoint operators. As we know, this establishes a link to Sturm’s classic derivation of Hermite
polynomials in 1836 [377], which was the first-ever study of the “singular” blow-up structure of multiple zeros of evolution PDEs. We will discuss the remarkable history behind this in Section 5.3.
