Dear Willie! Thank you very much! I already knew the representation formulas by I didn't know about their positivity. Now I understand what you mean. I learnt a lot from your comments! I'm really looking forward to seeing your lecture notes on PDE! I will mark my question as answered. I thought I have to wait for your last point.

Dear Willie! I forgot to ask a question regarding the fundamental solution that the solution to a nonlinear wave equation is $\geq$ solution to the linear wave equation. Where can I find a proof for that? And could you please tell me at which step John used this in his proof? I didn't realized that he used this fact. I would be appreciated for this!

I see. My understanding was that one cannot easily extend their result to our PDE. But maybe it could be generalized. I am not an expert. But it is strange that this PDE is not fully considered in the literature.

Thanks! So I'm trying to classify all possible regimes. So the problem (for the original PDE) is now small data with compact support that is equivalent to a PDE of "John's type". I made almost no progress to get an a priori estimate for the latter one. By the way, have you any comment on my two previous comments on lifespan? The upper bound on lifespan found in the paper I cited, is considerably better than that achieved by John (this is the same as the lower bound). Problem is that they are for $(\partial_t v)$ nonlinearity and not for $(\partial_t v)/(1+v)$

Dear Willie! Do I understand it correctly that for my original PDE (your first equation) there exists a result by Sideris 1984 (ref. [58] in your work) which says that if the initial data are large but compactly supported then the equation has no global $C^3$ solution?

I have found this paper: sciencedirect.com/science/article/abs/pii/… Prop.5.1 gives apparently the sharpest upper bound. But the problem is that it applies to the nonlinearity of the form $(\partial_t v)^2$. However, our PDE has the nonlinearity $(\partial_t v)^2/(1+v)$.

Thanks! But I have two questions: 1) how can we apply his results to our transformed PDE? Since the nonlinearity is slightly different. 2) does it suffice to get estimates for spherical symmetric case? What can we say about the generic case based on the results for radial case?

Oh of course! That was a stupid question. The way understand John's theorem is that it does not tell anything about how and at which time the blow-up occur. Is that correct? For instance, can we determine the constant $M$? Thank you Willie for this conversation, I'm not familiar with hyperbolic theory too much but I've learnt a great deal from you! Do you have a lecture note on NLW?

Thanks a lot! I will have a look at it. One last question: I thought the blow-up for my PDE would occur at finite time, at least I thought the Theorem 2 of John's work would imply that. But you say that it is not clear. Why?

So the small-data global well-posedness may depend on the nonlinearity?

Great! I almost guessed that it should go this way, but you shed more light on this. I changed the title of my post. But I have some stupid related questions: is this type of wave equation well-posed? How can I argue that it is? Can one talk about strong hyperbolicity for this type of wave equation? Or is every inhom. wave equation strongly hyperbolic no matter what the nonlinearity is?

And if what I said is true then, if $\phi$ is bounded bellow globally, then there exists a global solution, otherwise we have blow-up. Do I understand it correctly? And if this is true, how can I determine this lower global bound for $\phi$ (or for $1+v$)?