Set invariance for differential inclusions applied to PDES?

Question

This question is somewhat related to this one that I posted a while back on MSE, but the context has slightly changed since then. My question here relates to the consequences of a result in Weinberger's invariant sets for PDEs paper I was asking about in that earlier post. Namely, a key takeaway of that paper is that under certain "well behaved-ness" assumptions for a system of PDEs, the characterization of invariance of a set in state space only depends on the behavior of the vector field generated by the source term $f\left(u,x,t\right)$. Specifically, that the dot product of the vector field with the outward normal to the set be less than zero, or visually pointing inward, which is the same as the condition for set invariance of ODEs. Unless I'm misunderstanding, a consequence of this fact is that if one identifies an invariant set in the ODE version of a dynamic system, then if you properly modify the ODE to make a PDE version that satisfies the paper conditions, then the set remains invariant in the PDE sense as well.

This got me thinking about how other nice properties of ODE system descriptions could be extended to the PDE case. For instance, Clarke's book on nonsmooth analysis gives an analogous criterion for strong invariance of sets under differential inclusions where one need only check if the dot product of the outward normal with the upper Hamiltonian is less than or equal to 0 for all points on the set boundary. My question is if it's possible to combine the above two results to make a statement about invariance of a set under a PDE differential inclusion? Say one of the form:

$\frac{\partial u}{\partial t} + \frac{\partial^2 u}{\partial x^2} \in f\left(u,x,t\right)$

That is, that we need only be concerned about the source term for the PDE, and not the differential terms. It seems reasonable, but in my short experience with PDEs, they always seem to end up more complicated than I expect. Thanks for any input.

Answer 1

Your question was essentially answered by the Ph.D. thesis of Dieter Bothe, Multivalued differential equations and graphs and applications, Univ. Paderborn, 1992, see also the comments in the proof of Theorem 2 of Ralf Bader, On the semilinear multi-valued flow under constraints and the periodic problem, Comment. Math. Univ. Carolinae 41 (2000), no. 4, 719-734.

Roughly speaking, D. Bothe gives for an equation $u'(t)\in Au(t)+F(t,u(t))$ with $A$ being the generator of a semigroup $U$ an invariance condition in terms of $F$ and an outward normal to the set with respect to $U$. In R. Bader's variant (which is actually a special case of Bothe's result), a condition for $F$ w.r.t. to the outward normal and the invariance under $U$ is assumed.

Note that the invariance result only states that for any initial value inside of the set, there are solutions remaining inside of the set. In the absense of uniqueness, there might be further solutions which leave the set.