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Degeneracy in finite time of 1D quasilinear wave equations Ⅱ

The first author is supported by Grant-in-Aid for Young Scientists Research (B), 16K17631.
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  • We consider the large time behavior of solutions to the following nonlinear wave equation: $\partial_{t}^2 u = c(u)^{2}\partial^2 _x u + λ c(u)c'(u)(\partial_x u)^2$ with the parameter $λ ∈ [0,2]$ . If $c(u(0,x))$ is bounded away from a positive constant, we can construct a local solution for smooth initial data. However, if $c(· )$ has a zero point, then $c(u(t,x))$ can be going to zero in finite time. When $c(u(t,x))$ is going to 0, the equation degenerates. We give a sufficient condition so that the equation with $0≤q λ < 2$ degenerates in finite time.

    Mathematics Subject Classification: Primary: 35L15, 35A01; Secondary: 35B44, 35L65.

    Citation:

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  • Figure 1.  the two characteristic curves on the $(x,t)$ plane

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