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

We study the Cauchy problem for the quasilinear wave equation $\partial^2 _t u = u^{2a} \partial^2_x u + F(u) u_x$ 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 $u(0,x) \geq c_0>0$ for some constant $c_0$ 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 spacial infinity. Namely we consider the local well-posedness under the assumption that $u(0,x)>0$ and $u(0,x) \rightarrow 0$ as $|x| \rightarrow \infty$. Furthermore, to prove the local well-posedness, we find that the so-called Levi condition appears. Our proof is based on the method of characteristic and the contraction mapping principle via weighted $L^\infty$ estimates.