In this paper we consider the Cauchy boundary value problem for the Kirchhoff equation. 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 the nonlinear term. 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.

Derivative loss for Kirchhoff equations with non-Lipschitz nonlinear term

GHISI, MARINA;GOBBINO, MASSIMO
2009-01-01

Abstract

In this paper we consider the Cauchy boundary value problem for the Kirchhoff equation. 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 the nonlinear term. 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.
2009
Ghisi, Marina; Gobbino, Massimo
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11568/131365
 Attenzione

Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo

Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 11
  • ???jsp.display-item.citation.isi??? 9
social impact