On the lifespan and the blowup mechanism of smooth solutions to a class of 2-D nonlinear wave equations with small initial data (Q2788686)

In this paper, the authors considered the lifespan and the blowup mechanism for smooth solutions to the 2-D nonlinear wave equation \(\partial_t^2u- \sum_{i=1}^2\partial_i(c_i^2(u)\partial_i u) =0\), where \(c_i(u)\in C^{\infty}(\mathbb R^n)\), \(c_i(0)\neq 0\), and \((c'_1(0))^2+(c'_2(0))^2\neq 0\). The equation arises from the pressure-gradient model in compressible fluid dynamics and also in nonlinear variational wave equations. Under the initial condition \((u(0,x), \partial_tu(0,x))=(\epsilon u_0(x)\), \(\epsilon u_1(x))\) with \(u_0(x)\), \(u_1(x)\in C_0^\infty (\mathbb R^2)\), and \(\epsilon>0\) is small, it is shown that the classical solution \(u(t,x)\) stops to be smooth at some finite time \(T_\epsilon\), of size \(\epsilon^{-2}\). Moreover, blowup occurs due to the formation of a singularity of the first-order derivatives \(\nabla_{t,x}u(t,x)\), \(\|\partial_t u(t,\cdot)\|_{L^2(\mathbb R^2)}\simeq (T_\epsilon -t)^{-1}\), while \(u(t,x)\) itself is continuous up to the blowup time \(T_\epsilon\).