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Studying some mathematical models I came across a simple-looking question that I do not know how to handle.

If we have the following problem:

$$\begin{cases} \dfrac{\partial Z}{\partial t}-\Delta Z=aZ-bZ^2 & (x,t)\in\Omega\times (0,\infty)\\ \dfrac{\partial Z}{\partial\nu}=0, & (x,t)\in \partial\Omega\times (0,\infty)\\ Z(x,0)=Z_0(x)\geq 0 & x\in\Omega \end{cases}$$

Let's say that $Z(t_0,x)=0$ for any $x\in\Omega$. How can we prove that $Z(t,\cdot)\equiv 0$ for any $0\leq t\leq t_0$? Or how can we prove that $Z_0=0$?

We can assume any regularity needed for $\Omega$ and $Z_0$.

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  • $\begingroup$ Doesn't this just follow from the strong maximum principle? $\endgroup$
    – Otis Chodosh
    Dec 20, 2022 at 19:28
  • $\begingroup$ I cannot see how. The strong maximum principle says that a minimum can be attained on the boundary, or the function is constant. How can we derive information about $Z_0$ from here? $\endgroup$
    – Bogdan
    Dec 20, 2022 at 19:37
  • $\begingroup$ @OtisChodosh Please, can you provide some details? $\endgroup$
    – Bogdan
    Dec 21, 2022 at 10:57

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