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Speck, Jared

Stable ODE-type blowup for some quasilinear wave equations with derivative-quadratic nonlinearities. (English) Zbl 1439.35094

Anal. PDE 13, No. 1, 93-146 (2020).
The author proves a constructive stable ODE-type blowup result for open sets of solutions to a family of quasilinear wave equations in three spatial dimensions featuring a Riccati-type derivative-quadratic semilinear term. For the equation it is assumed that the quasilinear terms satisfy certain structural assumptions, which in particular ensure that the “elliptic part” of the wave operator vanishes precisely at the singular points. The initial data are assumed to be compactly supported and small or large in \(L^{\infty}\), but the spatial derivatives must initially satisfy a nonlinear smallness condition compared to the time derivative. As a corollary of the main results, it is shown that there are quasilinear wave equations that exhibit two distinct kinds of blowup: the formation of shocks for one non-trivial set of data, and ODE-type blowup for another non-trivial set.
Reviewer: Dongbing Zha (Shanghai)

Cited in 3 Documents

MSC:

35B44 Blow-up in context of PDEs
35L72 Second-order quasilinear hyperbolic equations
35L67 Shocks and singularities for hyperbolic equations
35L15 Initial value problems for second-order hyperbolic equations

Keywords:

compactly supported initial data; relative smallness of spatial derivatives; three spatial dimensions
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\textit{J. Speck}, Anal. PDE 13, No. 1, 93--146 (2020; Zbl 1439.35094)
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