I am currently trying to understand a paper on wave maps on Lorentzian manifolds. The setup is the following:
Let $u: V \rightarrow M$, where $(V,g)$ is a Lorentzian manifold and $(M,h)$ is a Riemannian manifold. Also $V=S \times R$, where $S$ is an $n$-dimensional orientable smooth manifold, $R$ is 1-dimensional. $M$ is $d$-dimensional. $g$ is a Robertson-Walker metric $g=-dt^2+R^2(t)\sigma$. Now i $\partial_i u^A$ and $\partial_t u^A$ are mappings from $S \times R$ into $T^∗ V\otimes TM$.
$u$ is supposed to be a wave map, so $g.\nabla^2 u=0$.
Also, $D_i$, respectively $ D_t$ are defined as covariant derivative in the metrics $\sigma$ and pull back of $h$ of a mapping from $S$ into $T^∗ S\otimes TM$
Well now define
$y(t):= R^2(t)\int\limits_{S_t}( \vert D D_t u \vert +R^{-2} \vert D^2u \vert ^2) \mu_{\sigma}$
where $\mu_{\sigma}$ is some measure in the metric $\sigma$ and $S_t =S \times \{t\} \subset V$ submanifold.
Through computations one gets the inequality:
$y^{'} \leq CR^{-1} (C_{\sigma} e_0^{1/2} R_0 y^{1/2} + C_h C(\sigma)(e_0^{3/2} R_0^3+y^{3/2})y^{1/2})$
where $C, C_{\sigma}, C_h, C(\sigma)$ are some depending constants and $e_0$ comes from the definition
$e_t= \int\limits_{S_t} (\vert \partial_t u \vert^2 +R^{-2} \vert Du \vert^2) \mu_{\sigma}$
Now, the paper deals with solving this differential inequality. My problem is: as far as I see, $y$ is a function from $R$ into functions from $S$ to $\mathbb{R}$, correct? How do you solve such differential equalities/inequalities?
The paper says:
If you define $z=y^{1/2}$, then one can formulate the inequality as $z^{'} \leq (a+K_hz^3)R^{-1}$ where $a= C(C_{\sigma}R_0e_0^{1/2}+C_hC(\sigma)R_0^3e_0^{3/2}$ and $K_h=C C_h^C(\sigma)$. A solution for the initial value for $t=t_0$ , $b=z_0$ is such that $F(z)\equiv \int\limits_b^z \dfrac{d \xi}{a+ K_h \xi^3} = \int\limits_{t_0}^t R^{-1}( \tau) d \tau \equiv X(t)$, $z$ is given by the inverse function $F^{-1}(X)$.
I really can't see where this comes from. Is is a strategy of solving diff. equations on manifolds?
Thanks in advance for any help!