3
$\begingroup$

Consider a wave equation of the form $$ \partial_t^2u(t,x)-c(t)^2\partial_x^2u(t,x)=0, \quad (t,x)\in (0,1]\times \mathbb{R} $$ where the speed $c(t)$ is in $L^1([0,1]) \cap C^1((0,1])$. This would mean that the speed may entertain logarithmic singularity.

The Cauchy problem for the above wave operator is well-posed in $C^\infty$. The well-posedness is established via an energy estimate by Colombini et. al (Colombini, F.; Del Santo, D.; Kinoshita, T.: Well-posedness of the Cauchy problem for a hyperbolic equation with non-Lipschitz coefficients. Ann. Scoula Norm Sup. Pisa Cl. Sci. 1(2002) 327-358).

Will the operator satisfy the cone condition? Especially, what will be the domain of influence when initial conditions are given at $t=0$?

Can anyone suggest some reference for determining cone condition for wave equation with singular speed?

Improve this question
$\endgroup$
2
  • 1
    $\begingroup$ In the paper to which you refer, the well-posedness is proved via Fourier transform: you deal with the family of ODE, $\partial_t^2 v+c(t)^2\xi^2 v=f$. I guess that following the same route and the energy estimates, you will be able to prove finite speed of propagation, but you need to look up into the article. $\endgroup$
    – Bazin
    Commented Mar 30, 2021 at 10:11
  • $\begingroup$ I am interested in knowing what will be the optimal slope of the cone. Energy estimate just tells me that speed of propagation is finite. $\endgroup$
    – Rahul Raju Pattar
    Commented Mar 31, 2021 at 9:26

0

Reset to default

You must log in to answer this question.