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Rank 2 singular solutions for quasilinear wave equations. (English) Zbl 0971.35053

The author constructs rank 2 singular solutions for general quasilinear wave equations with analytic coefficients. This result is a continuation of his previous work on blow up of classical solutions of quasilinear equations or systems. It is considered a quasilinear equation \[ L(u)= \sum^{n+1}_{i,j=1} l_{ij}(\partial u) \partial^2_{ij}u+ c(\partial u)= 0, \] which is a generalization of a wave equation \(\partial^2_tu- c(\partial_tu)^2\Delta u=0\), where \(\partial u= (\nabla_xu,\partial_t u)\). He proved that under some conditions there exists a solution of \(L(u)= 0\) with the following properties; (i) \(u\) and \(\partial u\) are continuous in a neighbourhood \(V\) of \(M_0\in \{t\leq T\}\), (ii) for some \(C>0\), \[ {C^{-1}\over T-t}\leq|D^2u(\cdot, t)|_{L^\infty}\leq {C\over T-t}, \] (iii) the singular part of the matrix \(D^2u\) has rank 2; for instance, there exists a path \(\gamma(s)\in V\), \(\gamma(s)\to M_0\), \(s\to 0\), such that \(\lim_{s\to 0}(T- t)D^2u(\gamma(s))\) exists and has rank 2.

MSC:

35L70 Second-order nonlinear hyperbolic equations
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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