On the lifespan of and the blowup mechanism for smooth solutions to a class of 2-D nonlinear wave equations with small initial data

TL;DR

This paper investigates the lifespan and blowup mechanism of smooth solutions to a class of 2-D nonlinear wave equations with small initial data, revealing that solutions develop singularities in derivatives while remaining continuous.

Contribution

It provides a detailed analysis of the blowup mechanism and lifespan for solutions to 2-D nonlinear wave equations arising from fluid dynamics and variational models, with small initial data.

Findings

Solutions blow up in finite time due to derivative singularities.

The solution remains continuous up to the blowup time.

Blowup is characterized by the formation of singularities in first derivatives.

Abstract

This paper is concerned with the lifespan and the blowup mechanism for smooth solutions to the 2-D nonlinear wave equation $\p_t^2u-\ds\sum_{i=1}^2\p_i(c_i^2(u)\p_iu)$ $=0$, where $c_i(u)\in C^{\infty}(\Bbb R^n)$, $c_i(0)\neq 0$, and $(c_1'(0))^2+(c_2'(0))^2\neq 0$. This equation has an interesting physics background as it arises from the pressure-gradient model in compressible fluid dynamics and also in nonlinear variational wave equations. Under the initial condition $(u(0,x), \p_tu(0,x))=(\ve u_0(x), \ve u_1(x))$ with $u_0(x), u_1(x)\in C_0^{\infty}(\Bbb R^2)$, and $\ve>0$ is small, we will show that the classical solution $u(t,x)$ stops to be smooth at some finite time $T_{\ve}$. Moreover, blowup occurs due to the formation of a singularity of the first-order derivatives $\na_{t,x}u(t,x)$, while $u(t,x)$ itself is continuous up to the blowup time $T_{\ve}$.