Well-posedness of the initial-boundary value problem for 1D degenerate quasilinear wave equations

Abstract

In this paper, we study the local well-posedness of the initial-boundary value problem for the 1D quasilinear wave equation $$\partial^2 _t u = \partial_x ( u^{2a} \partial_x u) + F(u)u_x $$ with the zero Dirichlet data on boundaries and the Levi condition on $F$. The boundary degeneracy causes losses in the principal part of the equation, which results in previous results only obtaining the estimates with loss of regularity. To overcome the main difficulty, we introduce singular weight functions to derive the estimates of preserving the initial regularity for the solution and its derivatives. The local existence and uniqueness for the degenerate initial-boundary value problem are established by applyingthe method of characteristic, the contraction mapping principle, and $L^\infty$ estimate with singular weight functions.