Author:
Ghisi Marina,Gobbino Massimo
Abstract
Abstract
It is well-known that the classical hyperbolic Kirchhoff equation admits infinitely many simple modes, namely time-periodic solutions with only one Fourier component in the space variables. In this paper we assume that, for a suitable choice of the nonlinearity, there exists a heteroclinic connection between two simple modes with different frequencies. Under this assumption, we cook up a forced Kirchhoff equation that admits a solution that blows-up in finite time, despite the regularity and boundedness of the forcing term. The forcing term can be chosen with the maximal regularity that prevents the application of the classical global existence results in analytic and quasi-analytic classes.
Subject
Applied Mathematics,General Physics and Astronomy,Mathematical Physics,Statistical and Nonlinear Physics
Reference25 articles.
1. On the well-posedness of the Kirchhoff string;Arosio;Trans. Am. Math. Soc.,1996
2. Global solutions to the Cauchy problem for a nonlinear hyperbolic equation;Arosio,1984
3. Sur une classe d’équations fonctionnelles aux dérivées partielles;Bernstein;Izv. Akad. Nauk SSSR,1940
4. Unstable simple modes of the nonlinear string;Cazenave;Q. Appl. Math.,1996
5. Global solvability for the degenerate Kirchhoff equation with real analytic data;D’Ancona;Invent. Math.,1992