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Let $X$ be a Banach lattice. Consider the system $$y'(t)=Ay(t)+f(t,y(t)) \qquad \text{in } (0,T) , \qquad y(0)=y_0, \qquad (*)$$ where $T>0$, $A$ generates an analytic positive semigroup $S(t)$ on $X$ and $f$ is a locally Lipschitz function.

Question:

Let $\eta \in X$. Assuming that $y_0 \ge \eta$, what are some minimal conditions on $f$ to ensure that the solution $y$ of $(*)$ satisfies: $$y(t)\ge \eta, \qquad \forall t\in (0,T).$$ Is there any theory or theorem that answers such question in a general framework?

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    $\begingroup$ Are you interested to the case where $\eta \neq 0$? $\endgroup$ Mar 12, 2020 at 9:43
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    $\begingroup$ @GiorgioMetafune I think the condition $f(y) \ge 0$ is too much. For example, for $y'=f(y)$, $y(0)\ge 0$, I found that a necessary and sufficient condition to ensure $y(t)\ge 0$ is $f(0) \ge 0$. This can be generalized to the heat equation. See this. $\endgroup$
    – S. Euler
    Mar 12, 2020 at 20:08
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    $\begingroup$ True, I agree, but in both cases you can use subsolutions or the maximum principle. In general maybe it is difficult to find precise conditions. Probably other criteria come out when $A$ is given by a form, by the proof of Beurling-Deny conditions. $\endgroup$ Mar 12, 2020 at 21:47
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    $\begingroup$ Theorem 3.3 in the paper of Haraux that you linked suggests that the analogous condition on a general Banach lattice $X$ might be that the semigroup generated by $A$ be not only positive but also "positivity improving" and that $f$ be cross-positive in the sense that we have $\langle x', f(x)\rangle \ge 0$" whenever $0 \le x' \in X'$ and $0 \le x \in X$ with $\langle x', x \rangle = 0$. However, I don't know which precise technical conditions are needed to make all the details work. $\endgroup$ Mar 12, 2020 at 23:49
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    $\begingroup$ Anyway, note that Theorem 3.3 alone does not give a condition purely in terms of $A$ and $f$; it also requires to already know that the given trajectory is positive for small times. In the applications to heat equations in Sections 4 and 5 Haraux checks this by a perturbation argument together with lower estimates for the heat semigroup. $\endgroup$ Mar 12, 2020 at 23:50

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I find that the condition $A\eta+f(\eta) \ge 0$ is necessary and sufficient. Let us assume that $f$ is globally Lipschitz and depends only on $y$, so that there is no problem about global existence. Let us have in mind that $X$ is an $L^p$ space or a space of continuous functions.

1) Assume first that $\eta=0$. If we take $y_0=0$, then $y'(0)=f(0)$; if $f(0) <0$, then $y$ would be negative for small $t$ (pointwise in spaces of continuous functions, against a positive functional in $L^p$...).

2) If $f \ge 0$ everywhere, then $y \ge 0$ by the variation of constants formula.

3) If $f$ is positive only for positive $y$, then we modify $f$ to $g$ by setting $g(y)=f(0)$ for $y \le 0$ and $f=g$ for positive $y$. Then we solve the problem with $g$ instead of $f$ and we get a positive solution, by the previous argument. This solution is then the solution of the given problem, by uniqueness.

4) Assume now that $f(0) \ge 0$ and take $\lambda$ such that $\lambda y+f(y) \ge 0$ for $y \ge 0$. We rewrite the problem in the equivalent form $$ y'=(A-\lambda )y+(\lambda y+f(y)).$$ Since $A-\lambda$ generates a positive semigroup, as well, we may apply point 3) and $y \ge 0$.

5) The general case reduces to $\eta=0$ writing $y=\eta+z, y_0=\eta+z_0$. The equation for $z$ becomes $z'=Az+A\eta+f(\eta+z), z_0 \ge 0$ and the condition $A\eta+f(\eta) \ge 0$.

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    $\begingroup$ If I understand correctly, you are implicitly assuming that $f: X \to X$ is not a general mapping, but the composition with a function from $\mathbb{R}$ to $\mathbb{R}$ that you also denote by $f$ (which is why you assumed $X$ to be a function space). However, I think the OP is actually interested in general functions $f$, as indicated by the linked paper of Haraux. $\endgroup$ Mar 14, 2020 at 14:25
  • $\begingroup$ Yes true. I thought the question was on reaction diffusion problems. $\endgroup$ Mar 14, 2020 at 14:39
  • $\begingroup$ @GiorgioMetafune Thank you for the answer. For the scalar valued case (in $\mathbb{R}$), no problem with the existence of $\lambda$ s.t $\lambda y + f(y) \ge 0$. I'm stuck on this condition in general case, for example for product space of $L^2$ as mentioned by Jochen. $\endgroup$
    – S. Euler
    Mar 14, 2020 at 15:05
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    $\begingroup$ Yes $\eta \in D(A)$. For systems I do not know. In case of systems of ODE's $y'=f(y)$ one has simple criterion for the invariance of a set $D$ (the inner product of the outer normal and $f$ must be nonpositive on the boundary). For PDE's I do not know. I see that the question was another. Should I delete the answer or leave so that everybody can follow the discussion? $\endgroup$ Mar 14, 2020 at 16:06
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    $\begingroup$ I found your answer very useful. Maybe a generalization of the result you mentioned is this. It's a geometric condition of set invariance by flow. The problem I have now is how to link between this and the positivity assumption on $f$. $\endgroup$
    – S. Euler
    Mar 14, 2020 at 23:22

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