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

PERNAZZA, LUDOVICO
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.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11571/224646
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