The heat flow and existence of geodesic

Question

I'm reading Jost's Riemannian Geometry and Geometric Analysis and in section 1.6 during the proof of the first step in theorem 1.6.1 it says

Step 1 is a general result from the theory of partial differential equations which follows by linearizing the equation at t = 0 and applying the implicit function theorem in Banach spaces, see §A.3. Therefore, we shall not discuss this here any further.

In appendix A.3

What one can deduce from Theorem A.3.2, however, is the short time existence of solutions when the linearization of the differential operator satisfies the assumptions of that theorem. This follows by linearization and the implicit function theorem.

Could someone provide me a resource for a more detailed way to linearize equation 1.6.2 ($\Gamma_{jk}^i u_s^ju_s^k$)?

For reference, the theorem and related parts are

Theorem statement

First part

Development

Answer 1

Suppose first that the solution you are looking for lives in $\Bbb R^n$ and that you are given the system $$ \begin{cases} \dfrac{\partial u(x,t)}{\partial t} &= F(x,u(x,t)),\\ u(x,0) &= u_0(x). \end{cases} $$ Its linearization around $u_0$ ("at $t=0$") is the equation $$ \dfrac{\partial v(x,t)}{\partial t} = L(x) v(x,t), $$ where $L(x)$ is the linear operator (depending only on $x$) defined by $$ L(x) v(x) = \left.\dfrac{\partial F\left(x,u_0(x)+tv(x)\right)}{\partial t}\right|_{t=0}. $$ We can interpret this latter expression as the Fréchet derivative $L(x) = D_2F_{(x,u_0(x))}$. If you are not looking for a solution in $\Bbb R^n$ but in a manifold, this latter property is used to define the linearization.

In your explicit case, looking at $$ F^i(s,u) = \partial^2_{ss} u^i + \Gamma^i_{jk}(u)\partial_su^j\partial_su^k $$ gives the linearization $$ L^iv = \partial^2_{ss}v^i + \partial_{\ell}\Gamma^i_{jk}(u_0)v^{\ell} \partial_su^j_0\partial_su_0^k + \Gamma^i_{jk}(u_0)\partial_sv^j\partial_su_0^k + \Gamma^i_{jk}(u_0)\partial_su_0^j \partial_sv^k, $$ which is indeed linear in $v$.