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Is anything known about the finite speed of propagation of wave-like nonlinear PDE:

$$u_{tt} - \Delta \left(u^p\right) = 0$$

when say $p > 1$?

That is given initial data $u(x,0) = w_1(x)$ and $u_t(x,0) = w_2(x)$ for $w_1,w_2 \in C_c(R^n)$, if the solution is compactly supported as well? Do energy functions work for these PDEs as well like in the case of wave equation $p = 1$?

Any help/reference is appreciated, thank you.

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  • $\begingroup$ The answer is complicated and seems to depend on the lower order/semilinear terms. See my previous answer for an examination of the $p = 3$ case. For the related equation where instead of $\Delta (u^p)$ you use the $p$-Laplacian, there seems to be some existing work (see projecteuclid.org/euclid.hokmj/1285766660 and references to/from). $\endgroup$
    – Willie Wong
    Commented Jun 19, 2017 at 3:32
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    $\begingroup$ One thing that I didn't say in my previous answer was that the equation (NL2) which has the good energy estimate has an obvious variational formulation (the density for the action is $(u_t)^2 - (\nabla u^2)^2$), whereas the other two equations do not. Instead of $u_{tt} - \Delta (u^p)$, you have something like $u_{tt} - u^{q} \Delta u^{q+1} = 0$, I expect you can do what I did there to get energy-based proofs of finite speed. $\endgroup$
    – Willie Wong
    Commented Jun 19, 2017 at 3:39

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