Formation of singularities for a family of 1D quasilinear wave equations

TL;DR

This paper extends the understanding of finite-time blow-up solutions for a family of 1D nonlinear wave equations with a parameter, demonstrating blow-up for a broader range of parameters using new estimates and analyzing solution properties.

Contribution

The paper introduces a new $L^{2/\lambda}$ estimate to establish blow-up solutions for $\lambda ext{ in } (0,1]$, broadening previous results.

Findings

Blow-up solutions exist for $\lambda ext{ in } (0,1]$.

The paper discusses properties like H"older continuity of solutions.

Extension of blow-up results beyond previously known cases.

Abstract

We consider the blow-up of solutions to the following parameterized nonlinear wave equation: $ u_{tt} = c(u)^{2} u_{xx} + \lambda c(u)c'(u)( u_x)^2$ with the real parameter $\lambda$. In previous works, it was reported that there exist finite time blow-up solutions with $\lambda =1$ and $2$. However, the construction of a blow-up solution depends on the symmetric structure of the equation (e.g., the energy conservation law). In the present paper, we extend the blow-up result with $\lambda =1$ to the case with $\lambda \in (0,1]$ by using a new $L^{2/\lambda}$ estimate. Moreover, some properties for the blow-up solution including the H\"older continuity are also discussed.