Local solvability for a quasilinear wave equation with the far field degeneracy: 1D case

We study the Cauchy problem for the quasilinear wave equation $u_{tt} =(u^{2a} \partial_x u)_x + F(u)u_x$ with $a \geq 0$ and show a result for the local-in-time existence under new conditions. In the previous results, it is assumed that $u(0,x) \geq c_0 \gt 0$ for some constant c0 to prove the existence and the uniqueness. This assumption ensures that the equation does not degenerate. In this paper, we allow the equation to degenerate at spatial infinity. Namely we consider the local well-posedness under the assumption that $u(0,x) \gt 0$ and $u(0,x) \to 0$ as ${\lvert x \rvert} \to \infty$. Furthermore, to prove the local well-posedness, we find that the so-called Levi condition appears. Our proof is based on the method of characteristics and the contraction mapping principle via weighted $L^\infty$ estimates.

Keywords

quasilinear hyperbolic equation, first-order hyperbolic systems, Levi condition

2010 Mathematics Subject Classification

35L05, 35L60, 35L80

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The author’s research was supported by Grant-in-Aid for Young Scientists Research (B), No. 19K14573.

Received 7 September 2021

Received revised 20 April 2022

Accepted 22 April 2022

Published 27 December 2022