Derivative loss for Kirchhoff equations with non-Lipschitz nonlinear term

Ghisi, Marina; Gobbino, Massimo

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 8 (2009) no. 4, pp. 613-646.

In this paper we consider the Cauchy boundary value problem for the integro-differential equation $u_{tt}-m\left({\int }_{\Omega }^{}{\left|\nabla u\right|}^{2}dx\right)\Delta u=0\text{in}\Omega \times [0,T)$ with a continuous nonlinearity $m:[0,+\infty )\to [0,+\infty )$. It is well known that a local solution exists provided that the initial data are regular enough. The required regularity depends on the continuity modulus of $m$. In this paper we present some counterexamples in order to show that the regularity required in the existence results is sharp, at least if we want solutions with the same space regularity of initial data. In these examples we construct indeed local solutions which are regular at $t=0$, but exhibit an instantaneous (often infinite) derivative loss in the space variables.

[1] A. Arosio and S. Panizzi, On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc. 348 (1996), 305–330. | MR | Zbl

[2] A. Arosio and S. Spagnolo, Global solutions to the Cauchy problem for a nonlinear hyperbolic equation, In: “Nonlinear Partial Differential Equations and their Applications”, Collège de France seminar, Vol. VI (Paris, 1982/1983), Res. Notes in Math., Vol. 109, Pitman, Boston, MA, 1984, 1–26. | Zbl

[3] S. Bernstein, Sur une classe d’équations fonctionnelles aux dérivées partielles (Russian, French summary), Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR] 4 (1940), 17–26. | MR | Zbl

[4] F. Colombini, Quasianalytic and nonquasianalytic solutions for a class of weakly hyperbolic Cauchy problems, J. Differential Equations 241 (2007), 293–304. | MR | Zbl

[5] F. Colombini, E. De Giorgi and S. Spagnolo, Sur le équations hyperboliques avec des coefficients qui ne dépendent que du temp (French), Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 6 (1979), 511–559. | EuDML | Numdam | MR | Zbl

[6] F. Colombini and S. Spagnolo, An example of a weakly hyperbolic Cauchy problem not well posed in $C^{\infty }$, Acta Math. 148 (1982), 243–253. | MR | Zbl

[7] F. Colombini, E. Jannelli and S. Spagnolo, Well-posedness in the Gevrey classes of the Cauchy problem for a nonstrictly hyperbolic equation with coefficients depending on time, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 10 (1983), 291–312. | EuDML | Numdam | MR | Zbl

[8] P. D’Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math. 108 (1992), 247–262. | EuDML | MR | Zbl

[9] P. D’Ancona and S. Spagnolo, On an abstract weakly hyperbolic equation modelling the nonlinear vibrating string, In: “Developments in Partial Differential Equations and Applications to Mathematical Physics” (Ferrara, 1991), Plenum, New York, 1992, 27–32. | MR | Zbl

[10] P. D’Ancona and S. Spagnolo, A class of nonlinear hyperbolic problems with global solutions, Arch. Ration. Mech. Anal. 124 (1993), 201–219. | MR | Zbl

[11] M. Ghisi and M. Gobbino, Spectral gap global solutions for degenerate Kirchhoff equations, Nonlinear Anal. 71 (2009), 4115–4124. | MR | Zbl

[12] M. Ghisi and M. Gobbino, A uniqueness result for Kirchhoff equations with non-Lipschitz nonlinear term, Adv. Math. (2009). | DOI | MR | Zbl

[13] J. M. Greenberg and S. C. Hu, The initial value problem for a stretched string, Quart. Appl. Math. 38 (1980/81), 289–311. | MR | Zbl

[14] F. Hirosawa, Degenerate Kirchhoff equation in ultradifferentiable class, Nonlinear Anal., Ser. A: Theory Methods, 48 (2002), 77–94. | MR | Zbl

[15] F. Hirosawa, Global solvability for Kirchhoff equation in special classes of non-analytic functions, J. Differential Equations 230 (2006), 49–70. | MR | Zbl

[16] G. Kirchhoff, “Vorlesungen ober Mathematische Physik: Mechanik”, section 29.7, Teubner, Leipzig, 1876. | JFM

[17] R. Manfrin, On the global solvability of Kirchhoff equation for non-analytic initial data, J. Differential Equations 211 (2005), 38–60. | MR | Zbl

[18] R. Manfrin, Global solvability to the Kirchhoff equation for a new class of initial data, Port. Math. (N.S.) 59 (2002), 91–109. | EuDML | MR | Zbl

[19] K. Nishihara, On a global solution of some quasilinear hyperbolic equation, Tokyo J. Math. 7 (1984), 437–459. | MR | Zbl

[20] S. I. Pohozaev, On a class of quasilinear hyperbolic equations, Mat. Sb. (N.S.) 96(138) (1975), 152–166) (English transl.: Math. USSR Sbornik 25 (1975), 145–158). | MR

[21] S. I. Pohozaev, The Kirchhoff quasilinear hyperbolic equation, Differentsial’nye Uravneniya 21 (1985), 101–108 (English transl.: Differential Equations 21 (1985), 82–87). | MR

[22] M. Reed and B. Simon, “Methods of Modern Mathematical Physics, I: Functional Analysis”, Second edition, Academic Press, New York, 1980. | MR

[23] T. Yamazaki, Global solvability for the Kirchhoff equations in exterior domains of dimension three, J. Differential Equations 210 (2005), 290–316. | MR | Zbl

[24] T. Yamazaki, Global solvability for the Kirchhoff equations in exterior domains of dimension larger than three, Math. Methods Appl. Sci. 27 (2004), 1893–1916. | MR | Zbl