Timeline for Finite speed of propagation for $u_{tt} - \Delta (u^p) = 0$
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| Jun 19, 2017 at 3:39 | comment | added | Willie Wong | 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. | |
| Jun 19, 2017 at 3:32 | comment | added | Willie Wong | 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). | |
| Jun 19, 2017 at 3:26 | history | edited | Willie Wong |
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| Jun 18, 2017 at 23:13 | history | asked | r9m | CC BY-SA 3.0 |