Timeline for On a nonlinear wave equation
Current License: CC BY-SA 4.0
27 events
| when toggle format | what | by | license | comment | |
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| Apr 19, 2022 at 19:23 | vote | accept | Immanuel | ||
| Apr 19, 2022 at 19:22 | comment | added | Immanuel | 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. | |
| Apr 19, 2022 at 16:19 | comment | added | Willie Wong | For a more striking (and more explicit) use of the positivity, see F. John, Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Math. 28, 235-268 (1979). | |
| Apr 19, 2022 at 16:18 | comment | added | Willie Wong | @Immanuel the fundamental solution for the linear wave equation has explicit formulae in dimensions 1, 2, and 3 (See e.g. en.wikipedia.org/wiki/Wave_equation combined with en.wikipedia.org/wiki/Duhamel%27s_principle ; this is also in most intro PDE textbooks). By inspection it is positive. (This doesn't work in higher dimensions.) // John used it in many spots implicitly where he appealed directly to the fundamental solution (after reducing things to spherical symmetry). | |
| Apr 19, 2022 at 15:56 | comment | added | Immanuel | 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! | |
| Apr 18, 2022 at 16:08 | comment | added | Immanuel | 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. | |
| Apr 18, 2022 at 14:10 | comment | added | Willie Wong | I don't have much to say about your comments on lifespan, since i hadn't had a chance to look over the paper you linked to. The non-radial case certainly may behave differently from the radial case, though in the small data regime the expectation is that the purely radial "mode" decays the slowest and so should consequently be the primary driver of blow-up. This is verified to be the case for quasilinear equations. For semilinear equations I don't know for sure. | |
| Apr 18, 2022 at 14:01 | comment | added | Immanuel | 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 $(\partial_t v)/(1+v)$ | |
| Apr 18, 2022 at 13:35 | comment | added | Willie Wong | Yes. (You understand correctly.) The big difference however is that John's original theorem makes no assumptions on the size of the initial data, other than that it is nonzero. | |
| Apr 18, 2022 at 10:34 | comment | added | Immanuel | 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? | |
| Apr 14, 2022 at 16:31 | comment | added | Immanuel | 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 $(\partial_t v)^2/(1+v)$. | |
| Apr 14, 2022 at 15:58 | comment | added | Immanuel | 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? | |
| Apr 14, 2022 at 13:20 | comment | added | Willie Wong | (I vaguely remember seeing sharpened lifespan estimates in the literature, but I can't remember the authors. Try doing a reverse search on MathSciNet from John's paper.) Regarding lecture notes: that's on my todo list. | |
| Apr 14, 2022 at 13:15 | comment | added | Willie Wong | @Immanuel John's 1981 paper actually provides also life-span estimates in the small-data limit. If the initial data is size $\epsilon$, Theorem 3 guarantees (in the radial case) solutions exists up to time $T \approx e^{C/\epsilon}$ for some constant $C$. The discussion in section 3 shows that the time of existence cannot exceed $T \approx e^{C'/\epsilon^4}$. The upper bound however depends on the profile of the initial data, and so cannot easily be used to extend Theorem 2. Additionally: the estimates are clearly not sharp (due to the differing powers of $\epsilon$ between the two bounds.) | |
| Apr 14, 2022 at 6:17 | comment | added | Immanuel | 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? | |
| Apr 14, 2022 at 2:43 | comment | added | Willie Wong | When John's theorem can be applied: yes, the blow up happens in finite time. But if $v$ is unbounded from above (say, has potentially linear growth in time), then in the equations deduced in the answer I wrote above, John's theorem cannot be applied. | |
| Apr 13, 2022 at 22:31 | comment | added | Immanuel | 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? | |
| Apr 13, 2022 at 22:25 | comment | added | Willie Wong | Yes, in spatial dimension $\leq 3$. See e.g. my paper arxiv.org/abs/1407.6276v1 for a survey of known results as of 8 years ago. The paper was written with focus on the quasilinear case, as that is where the blow-up results are more robustly proven. For the case of $\Box \phi = g(\partial\phi)$, the small data global well posedness results are basically the same as that for the quasilinear case. | |
| Apr 13, 2022 at 22:22 | comment | added | Immanuel | So the small-data global well-posedness may depend on the nonlinearity? | |
| Apr 13, 2022 at 22:20 | comment | added | Willie Wong | local well-posedness is immediate. See e.g. Sogge's book. Of interest is small data global well-posedness. // Hyperbolicity only concerns the principal part of the equation. Your equation is semilinear so the principal part is just the standard linear wave operator, so you have hyperbolicity. You can run into problems if your nonlinearity depends on the second derivative of $\phi$, as in this case the principal symbol may be changed. | |
| Apr 13, 2022 at 22:19 | comment | added | Immanuel | 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? | |
| Apr 13, 2022 at 22:15 | comment | added | Willie Wong | @Immanuel: all typos are now fixed (which should answer all of your three comments above). I need to think more about how (and whether we can) rule out blow-up at infinity. | |
| Apr 13, 2022 at 22:09 | history | edited | Willie Wong | CC BY-SA 4.0 |
Fixed some typos.
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| Apr 13, 2022 at 22:06 | comment | added | Immanuel | 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 $1+v$)? | |
| Apr 13, 2022 at 22:01 | comment | added | Immanuel | Than you Willie! I think there are some typos. On the r.h.s you should have $(\partial_t \phi)^2$. Then the r.h.s. by transformation is equal to $(\partial_t v)^2/(1+v)$. Therefore, $1/(1+v)$ should be bounded bellow globally or equivalently $1+v$ should be bounded above by some positive number. | |
| Apr 13, 2022 at 21:43 | comment | added | Immanuel | In the actual problem the constant $\alpha$ is such that I could, using similar transformation but with different factor, arrive at $\Box v = (\partial_t v)^2/v$ for $v>0$. Then, based on Theorem 2, I said that there exists some constant $a$ such that if $v \leq 1/a^2$ then the theorem applies. I think that you are saying the same thing, right? | |
| Apr 13, 2022 at 21:05 | history | answered | Willie Wong | CC BY-SA 4.0 |