$u_{tt}-c^2u_{xx}=0\ for\ t>0,\ x\in R$ and
$ u(0,x)=u_{0}(x),\ u_{t}(0,x)=u_{1}(x)$
Suppose that $u_{0}\in C^2(R)\ and\ u_{1}\in C^1(R) $ satisfy the inequalities
$m|x|^a\le u_{0}(x)\le M|x|^a ,\ |x|^{a-1}\le u_{1}(x)\le M|x|^{a-1} $
So i have to show that for every $x_{0}\in R$ there exists consatnts $t_{0}, C_{1}, C_{2}$ such that
$C_{1}t^a\le u(t,x_{0})\le C_{2}t^a \ $ for all $t\ge t_{0}$
So i start with the wave eq. solution
$|u(t,x)|\leq(\frac{1}{2})|u_{0}(x-ct)+u_{0}(x+ct)|+\frac12 |\int^{(x+ct)}_{(x-ct)}u_{1}(s)ds|$
and put the inequalities in it but but i couldn't find what i want
Can someone help me , please