Manifold of mappings between $M$ and $N$, with non-compact source $M$ EDIT: Let $M$ and $N$ are two smooth manifold and suppose $N$ is compact but $M$ is not necessarily compact. For my purpose, I just need to consider the case $M=\mathbb R \times S^1$ or $\mathbb R \times [0,1]$.
Thanks to David's comments that help me a lot, the problem I really concern is just the following.
Question: 


*

*Now that we know if $M$ is non-compact then $C^\infty(M,N)$ is
modeled on spaces $C^\infty_c(M,N)$. Since we know
$W^{k,p}_0(\Omega)$ is the closure of $C^\infty_c(\Omega) $ in
$W^{k,p}(\Omega)$ for $\Omega\subset \mathbb R$, in analogy, can we
say $W^{k,p}(M,N)$ is modeled on spaces $W^{k,p}_0(M,f^*TN)$? Any reference?

*Can we consider $W^{k,p}_0(M,f^*TN)$ as a completion (in some
sense?) of $C^\infty_c(M,f^*TN)$, the spaces of smooth sections with compact support of pullback bundles along $f:M\to N$?



Previous post:
(1) Can we define $W^{1,p}_0(M,N)$ 
 in a similar manner that $W^{1,p}_0(\mathbb R)$ is defined by the completions of $C^\infty_0$ or $C^\infty_c$?


*

*For example, Floer in his paper (see Definition 2.1) actually
discusses (roughly) $\mathcal W:=W_{loc}^{k,p}(\mathbb R \times
   [0,1], N)$ with a topology given by open sets as follows: $ \mathcal
   O_{u,\rho,\epsilon} =\{ v\in \mathcal W \mid v= \exp_u \xi
   ~\text{on}~ [-\rho,\rho]\times[0,1], ~\text{and}~ ||\xi||_{W^{k,p}}
   <\epsilon  \} $ On the other hand, Audin and Damian in their
book (see Definition 8.2.2) consider Banach manifolds $\widetilde
   {\mathcal W} =W^{k,p}(\mathbb R \times [0,1], N)$ (actually $[0,1]$
should be replaced by $S^1$) in a quite different manner. Here open
sets are the space of maps of the form $v=\exp_u \xi$ where $\xi \in
   W^{k,p}(\mathbb R \times [0,1], N)\equiv W_0^{k,p}(\mathbb R \times
   [0,1], N)$ and where $u$ is smooth and converges in some decay at the
infinity.
Heuristically, $\mathcal W$ is like the completion of $C_c^\infty$
while $\widetilde {\mathcal W}$ is like that of $C_0^\infty$.
(2) Is $\mathcal W$ the same as $\widetilde{\mathcal W}$? Which one could be a better candidate for the definition of $W^{k,p}_0(M,N)$? Are they both Banach manifolds as in the case $M$ is compact?


*

*Recently I notice that when considering (infinite-dimensional)
manifolds of mapping, say $C^\infty(M,N)$, we usually require $M$ to
be compact. (See Section 4.2 of this paper: The inverse function
theorem of Nash and Moser). Notice that as long as the domain $M$
is compact, the questions (1), (2) and (3) become trivial.


(3) In general, if $M$ is non-compact, then is $C^\infty(M,N)$ still a Frechet manifold as in the case $M$ is compact?

 A: Let $M$ and $N$ be Riemannian manifolds.
In general, the space of Sobolev mappings $W^{k,p}(M,N)$ should not be defined as a completion of smooth mappings even if $k=1$ and manifolds are compact. The common definition (at least if $N$ is compact) is as follows. Take an isometric embedding of $N$ into a Euclidean space $\mathbb{R^\nu}$ and then define:
$$
W^{k,p}(M,N)=\{ f\in W^{m,p}(M,\mathbb{R}^\nu):\, f(x)\in N \text{ a.e.} \}.
$$
This space is equipped with the metric inherited from the Sobolev norm and in general smooth mappings are not dense [1]. There are some problems when $N$ is not compact. You can find more papers about higher order Sobolev mappings between manifolds, including the case of non-compact target, at the homepage of 
Van Schaftingen.
[1] P. Bousquet, A. C. Ponce, J. Van Schaftingen, Strong density for higher order Sobolev spaces into compact manifolds. J. Eur. Math. Soc. (JEMS) 17 (2015), no. 4, 763–817.
A: Too long for a comment: If $M$ is a manifold, you have no volume form, and you cannot give a meaning to integrability of a function. To express integrability, you have to use densities, which are smooth sections of the fiber bundle of densities over the manifold $M$. Note that you don't have this difficulty with your example, since you have then  the form $dx\wedge d\theta$ on $\mathbb R\times\mathbb R
/2π\mathbb Z$.
