I am considering the following wave equation (for $\phi=\phi(x,t)$) $$ \phi_{tt} - \Delta \phi = a |\nabla \phi|^2, \quad (x,t) \in \mathbb{R}^3 \times \mathbb{R} $$ where $\nabla$ is just spatial gradient, i.e., $\nabla \phi= (\partial_{x_1} \phi, \partial_{x_2} \phi, \partial_{x_3} \phi)$ and $a \in \mathbb{R}$. And assume that initial data $\phi(x,0)$ and $\partial_t \phi(x,0)$ are regular enough and compactly supported.

This PDE in some sense is the complementary case of the PDE in John's blow-up theorem, where we have $$ \phi_{tt} - \Delta \phi = (\partial_t\phi)^2. $$

Now, I have the following questions: is this PDE well-posed? Or the solution will blow-up? Is there any paper on this PDE in 3+1 dimensions?


1 Answer 1


By replacing $\phi$ by $-\phi$ you can set $\alpha$ to be positive (or negative, if you wish). By replacing $\phi$ by $\lambda \phi$ you can rescale away $\alpha$. So you can set $\alpha$ to be either $+1$ or $-1$ as you wish.

So for now let's consider the situation you are solving $$ \partial^2_{tt}\phi - \Delta \phi = |\nabla \phi|^2 $$ First observation: suppose $\phi$ is a $C^3$ solution with compactly supported initial data, then $\phi$ is uniformly bounded from below on $\mathbb{R}^3\times [0,\infty)$. This is due to the fact that on $\mathbb{R}^3$ the fundamental solution to the wave equation is signed (a fact also used by John in his proof), so that the solution to the above equation is pointwise $\geq$ the solution to the linear equation with the same initial data. The latter is globally uniformly bounded, and hence $\phi$ is uniformly bounded below.

Now $v = e^\phi - 1$; you have that $v$ is also $C^3$, and there exists $\beta$ such that $v + 1 > \beta$ uniformly on $\mathbb{R}^3\times [0\infty)$, using the uniform bound on $\phi$.

Now we compute $$ (\partial^2_{tt} - \Delta) v = e^\phi (\partial^2_{tt} - \Delta) \phi + e^\phi (\partial^2_t \phi - |\nabla\phi|^2) = \frac{1}{1+v}( \partial_t v)^2 $$

The lower bound above ensures that the denominator on the RHS is never zero.

Suppose now that $\phi$ is globally bounded above also: then $(1+v)$ is bounded above globally by some $M > 0$. Then the condition for Theorem 2 in John's 1981 paper is satisfied and the result follows.

So the remaining question is whether it is possible for $\phi$ to blow-up only at infinity and not at any finite time. I think there may be a way to rule this out, but that would require checking some more technical details so I'll have to report back later.

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    $\begingroup$ @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. $\endgroup$ Apr 13, 2022 at 22:15
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    $\begingroup$ 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. $\endgroup$ Apr 13, 2022 at 22:20
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    $\begingroup$ 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. $\endgroup$ Apr 13, 2022 at 22:25
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    $\begingroup$ 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. $\endgroup$ Apr 14, 2022 at 2:43
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    $\begingroup$ @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.) $\endgroup$ Apr 14, 2022 at 13:15

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