This question is a bit informally formulated and loosely stated, but I hope someone can make sense of it and guide me in some way. With my limited experience in PDEs, I have always seen the time variable being treated differently than the space variables.

**A bit of background:** As I'm studying a differential equation, 2nd order in space, 1st order in time, whose solution is a function $u=u(\mathbf{x},t)$ where $\mathbf{x}=(x,y)\in \Omega\subset\mathbb{R}^2$ where $\Omega=[a,b]\times[c,d]$ is a rectangular domain and $t\in [0,\infty)$. Using common reasoning in functional analysis, I was looking for solutions in Bochner spaces like $H^1((0,\infty),H^2(\Omega))$.

However, a certain change of variables $z=y-t $ and $w=x$ (I'm looking for traveling waves) reduced the original equation to the following $$-\Delta \phi(w,z)+\phi(w,z)-w=F(\phi(w,z)),$$ where $u(x,y,t)=\phi(w,z)$ and $F$ is an arbitrary (to be determined) function. In the $(w,z)$ variables, the new domain $\tilde{\Omega}=[a,b]\times (\infty,b]$ in question is an infinite semi strip containing $\Omega$ and $\phi$ inherits whichever boundary conditions $u$ satisfies. So now I realized in need to look for solutions in a new space $H(\tilde{\Omega})$ where $H$ is an appropriate function space.

**Bottom line:** To study the behavior of my solution $u(x,y,t)$ in time $t$ on a bounded domain $\Omega$, I now study the asymptotic behavior of $\phi(w,z)$ in the new "spatial variable" $z=y-t$.

**The question:** Is there an advantage of studying time $t$ as a spacial variable $w=y-t$?

If the question doesn't make any sense please let me know in the comment so that I make needed clarifications.

* Added:* Thank you for the very helpful comments. However, I should have emphasized that I'm interested in the nature of our new function space $H(\tilde{\Omega})$. Do I get an advantage in finding solutions in $H(\tilde{\Omega})$ vs $H^1((0,\infty),H^2(\Omega))$? In which space is time behavior easier to study?