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I have a few general questions related to PDE solution behavior, specifically as it relates to set invariance. Namely, I've been reading papers that give necessary/sufficient conditions for set invariance of parabolic PDE systems, and have noticed that the results only show invariance of a set on the time interval $t\in\left(0,T\right)$ for some finite $T$. There's also restrictions on the boundedness of the geometry, $D \subseteq \mathbb{R}^n$, but for the purposes of our project, we aren't considering unbounded geometries.

In brief, these theorems state that given some nice properties of the PDE system (Lipschitz continuity of the source term, Holder continuity of the invariant set boundaries, constraints on the boundary conditions, etc) a subset $A$ of the solution space $\mathbb{R}^m$ is invariant over the geometry and on the time domain $t\in\left(0,T\right)$ with respect to a solution of the PDE $u\left(x,t\right)$ if a dot product criterion on the source term, $f\left(x,t\right)$, is met at all points on the boundary $\partial A$.

My question is why the restriction to a finite time horizon on the solution? Very roughly speaking, if you interpret the conditions for invariance to say that "if the PDE is well behaved enough over this interval" then you have invariance, what can cause the PDE to stop behaving well at larger times? If the source term were independent of time, it would seem to me we could amend the statement for times going to infinity?

If you could offer any insight or examples here, that would be great. Getting some more intuition to wrap my head around why were concerned about behavior at large times would be very helpful.

Leif Ericson
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1 Answers1

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I have a hard time reading your question. What do you mean precisely with the solution space and what does invariance precisely mean. Could you maybe give some references to the papers you mention?

Now, I do not know the precise setting you are working with, but I do see a parallel with well-posedness theory of reaction-diffusion equations, say $$ u_t=\mathrm{div} A(\,\cdot\,,u,\nabla u) + f. $$ Often it is shown that solutions exist for all times $t\in[0,T]$ for some $0<T<\infty$. Such proofs use uniform energy estimates and their uniforms bound rely on $T$ being finite. In particular the expression $\int_0^T\int_{\mathbb{R}^n}f^2$ has to be bounded. If $f$ does not depend on $t$, this might not be the case for $T=\infty$. Of course, asking that $f\in L^2(\mathbb{R}^n\times \mathbb{R})$ solves the issue, but this means that dependence on time of the source term is needed.

SilverBlueZ
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  • Thanks for the answer! I'll try to clarify a few points. I probably could have used better terminology than "solution space," but by that I just mean the space where a solution, $u\left(x,t\right)$ lies, say $\mathbb{R}^m$. By invariant I mean that if $u\left(x,t\right)$ lies in some closed subset of $S \subset \mathbb{R}^m$ at some initial time at all points on the geometry, then $u\left(x,t\right)$ will remain in $S \quad \forall \left(x,t\right) \in DX\left[0,T\right]$ – Leif Ericson Oct 18 '21 at 18:30
  • Thank you for your clarification. I personally would use the terminology "$u$ takes values in $\mathbb{R}^m$" instead, because the term 'solution space' is often reserved for the functional space in which solutions of the PDE lie. – SilverBlueZ Oct 20 '21 at 07:51
  • Gotcha, I'll keep that in mind going forward. Now that I'm reading your answer again, one question did come up that I wanted to add. When you say the theory on solution existence relies on $\int_{0}^{T} \int_{\mathbb{R}^n} f^2,$ being bounded, do you mean it has to be bounded over all $\mathbb{R}^n$, or is just on the geometry over which the PDE is solved sufficient? Also, would it be incorrect here to say that $f\in L^2$ as would imply $T$ can run to infinity? Also you asked which papers were looking at. The main is: – Leif Ericson Oct 20 '21 at 14:52
  • https://www.semanticscholar.org/paper/Invariant-sets-for-weakly-coupld-parabolic-and-efli-Weinberger-Mrer/2b2a9279252d9f4ae4c64adcb366e2b3ef5ff5dc Ran out of characters there. Thanks again for the reply! – Leif Ericson Oct 20 '21 at 14:53
  • Thank you for the reference. I had a quick look at it and noticed that the result on this invariance set is actually a generalization of the maximum principle I was not familiar with. So thanks, I learned something new.

    I do not think $S$ depends on $T$, so if the hypotheses of the theorem (Theorem 1 on page 299) are satisfied for any $T$, then the theorem implies that $u(x,t)\in S$ for any $(x,t)\in D\times (0,T]$, for any $T>0$ (and therefore for any $(x,t)\in D\times \mathbb{R}$).

     – SilverBlueZ Dec 21 '21 at 12:21