Blow-up analysis for two kinds of nonlinear wave equations

In this paper, we discuss the blow-up and lifespan phenomenon for the following wave equation with variable coefficient: $$ u_{tt}(t,x)-\mathbf{div}\bigl(a(x)\mathbf{grad}u(t,x) \bigr)=f(u,Du,D_{x}Du), \quad x \in \mathbf{R}^{n}, t>0, $$ with small initial data, where $a(x)>0$, $Du=(u_{x_{0}},u_{x_{1}},\ldots ,u_{x_{n}})$ and $D_{x}Du=(u_{x_{k}x_{l}}, k,l=0,1,\ldots ,n, k+l\geq 1)$. Then we find a new phenomenon. The Cauchy problem $$ u_{tt}(t,x)-\triangle u(t,x)=u(t,x)e^{u(t,x)^{2}}, \quad x\in \mathbf{R}^{2}, t>0, $$ is globally well-posed for small initial data, while for the combined nonlinearities $$ u_{tt}(t,x)-\triangle u(t,x)=u(t,x) \bigl(e^{u(t,x)^{2}}+e^{u_{t}(t,x)^{2}} \bigr), \quad x \in \mathbf{R}^{2}, t>0 $$ with small initial data will blow up in finite time. Moreover, we obtain the lifespan results for the above problems.