So i have a Cauchy problem based on this equation above: $$ u_{ttxx}=u_{tt}^2\\ u|_{t=0}=\phi_1(x)\\ u_t|_{t=0}=\phi_2(x)\\ u_{tt}|_{t=0}=\phi_3(x)\\ u_{ttt}|_{t=0}=\phi_4(x)$$
My idea was to substitute $u_{tt}=w$ and solve a non-linear DE $w_{xx}=w^2$. But in this case i would have a non-trivial integral which i don't know how to crack: $\int \sqrt{(C1+\frac{2}{3}w^3)}^{-1}dw=C2+x$.
Maybe there is another way. Also, I've found the characteristics of the base equation: $x-t; \; x+t;\;x$, and $x$ seems to be 'multiple' or 'double' characteristic, but i didn't hear of multiple characteristics ever. But this may help, i suppose.
Thanks.
upd: the Professor told me those characteristics i found were all wrong. maybe the correct ones are $x=const$ and $t=const$, both multiple. Anyone ever had business with multiple characteristicss?