# Positive solutions for semilinear parabolic equations

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?

• Are you interested to the case where $\eta \neq 0$? Mar 12, 2020 at 9:43
• @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. Mar 12, 2020 at 20:08
• 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. Mar 12, 2020 at 21:47
• 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. Mar 12, 2020 at 23:49
• 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. Mar 12, 2020 at 23:50

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 $$A\eta+f(\eta) \ge 0$$.

• 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. Mar 14, 2020 at 14:25
• Yes true. I thought the question was on reaction diffusion problems. Mar 14, 2020 at 14:39
• @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. Mar 14, 2020 at 15:05
• 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? Mar 14, 2020 at 16:06
• 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 $f$. Mar 14, 2020 at 23:22