In this paper we discuss the $C^{\infty}$ well-posedness for second order hyperbolic equations $Pu=\partial_t^2u-a(t,x)\partial_x^2u=f$ with two independent variables $(t,x)$. Assuming that the $C^{\infty}$ function $a(t,x)\geq 0$ verifies $\partial_t^pa(0,0)\neq 0$ with some $p$ and that the discriminant $\Delta(x)$ of $a(t,x)$ vanishes of finite order at $x=0$, we prove that the Cauchy problem for $P$ is $C^{\infty}$ well-posed in a neighbourhood of the origin.

Some well-posed Cauchy problem for second order hyperbolic equations with two independent variables

COLOMBINI, FERRUCCIO;
2011-01-01

Abstract

In this paper we discuss the $C^{\infty}$ well-posedness for second order hyperbolic equations $Pu=\partial_t^2u-a(t,x)\partial_x^2u=f$ with two independent variables $(t,x)$. Assuming that the $C^{\infty}$ function $a(t,x)\geq 0$ verifies $\partial_t^pa(0,0)\neq 0$ with some $p$ and that the discriminant $\Delta(x)$ of $a(t,x)$ vanishes of finite order at $x=0$, we prove that the Cauchy problem for $P$ is $C^{\infty}$ well-posed in a neighbourhood of the origin.
2011
Colombini, Ferruccio; Nishitani, T; Orrù, N; Pernazza, L.
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/11568/144959
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 0
  • ???jsp.display-item.citation.isi??? 1
social impact