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I was wondering if the solution of a wave equation is real analytic when the coefficients, Cauchy data, and boundary data is real analytic. To be precise, consider two types of problems: \

An initial value problem $$ u_{tt} - c^2(x)\Delta u = F(t,x) \text{ in } \mathbb{R}^n, \quad\quad u(0,x)=f_1(x), \quad u_t(0,x)=f_2(x) $$ where $c(x), F(t,x), f_1(x), f_2(x)$ are real analytic functions.

Q1: Is the solution $u(t,x)$ a real analytic function in both $t$ and $x$?

An initial boundary value problem $$ u_{tt} - c^2(x)\Delta u = F(t,x) \quad\text{ in } (0,1)\times B_1, $$ $$ u(0,x)=f_1(x), \quad u_t(0,x)=f_2(x), \quad u|_{(0,1)\times\partial B_1} = g(t,x). $$ where $B_1$ is the unit ball; $c(x), F(t,x), f_1(x), f_2(x), g(t,x)$ are all real analytic functions.

Q2: Is the solution $u(t,x)$ a real analytic function in both $t$ and $x$?

I think the first answer is yes and it is just a consequence of the Cauchy-Kovalewskaya theorem. For the second question, I am not sure if it is true. Any idea or reference would be greatly appreciated. Thank you in advance!

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