ap.evaluation of pdes – When is it helpful to deal the time variable in a evolution downside as a spatial variable? Answer

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ap.evaluation of pdes – When is it helpful to deal the time variable in a evolution downside as a spatial variable?

This query is a bit informally formulated and loosely said, however I await somebody could make sense of it and lead me in some route. With my restricted suffer in PDEs, I’ve at all times seen the time variable being handled in a different way than the house variables.

A little bit of background: As I’m learning a differential equation, 2nd organize in house, 1st organize in time, whose resolution is a duty $$u=u(mathbf{x},t)$$ the place $$mathbf{x}=(x,y)in Omegasubsetmathbb{R}^2$$ the place $$Omega=[a,b]instances[c,d]$$ is an oblong province and $$tin [0,infty)$$. 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 $$tilde{Omega}=[a,b]instances (infty,b]$$ in query is an innumerable semi divest containing $$Omega$$ and $$phi$$ inherits whichever border situations $$u$$ satisfies. So now I spotted in necessity to search for options in a fresh house $$H(tilde{Omega})$$ the place $$H$$ is an arrogate duty house.

Bottom line: To seek the conduct of my resolution $$u(x,y,t)$$ in time $$t$$ on a bounded province $$Omega$$, I now seek the asymptotic conduct of $$phi(w,z)$$ within the fresh “spatial variable” $$z=y-t$$.

The query: Is there an edge of learning time $$t$$ as a spacial variable $$w=y-t$$?

If the query does not make any sense delight let me know within the observation in order that I make wanted clarifications.

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