Blow-up analysis for two kinds of nonlinear wave equations

Abstract

AbstractIn this paper, we discuss the blow-up and lifespan phenomenon for the following wave equation with variable coefficient: $$ u_{tt}(t,x)-\mathbf{div}\bigl(a(x)\mathbf{grad}u(t,x) \bigr)=f(u,Du,D_{x}Du), \quad x \in \mathbf{R}^{n}, t>0, $$utt(t,x)−div(a(x)gradu(t,x))=f(u,Du,DxDu),x∈Rn,t>0, with small initial data, where $a(x)>0$a(x)>0, $Du=(u_{x_{0}},u_{x_{1}},\ldots ,u_{x_{n}})$Du=(ux0,ux1,…,uxn) and $D_{x}Du=(u_{x_{k}x_{l}}, k,l=0,1,\ldots ,n, k+l\geq 1)$DxDu=(uxkxl,k,l=0,1,…,n,k+l≥1).Then we find a new phenomenon. The Cauchy problem $$ u_{tt}(t,x)-\triangle u(t,x)=u(t,x)e^{u(t,x)^{2}}, \quad x\in \mathbf{R}^{2}, t>0, $$utt(t,x)−△u(t,x)=u(t,x)eu(t,x)2,x∈R2,t>0, is globally well-posed for small initial data, while for the combined nonlinearities $$ u_{tt}(t,x)-\triangle u(t,x)=u(t,x) \bigl(e^{u(t,x)^{2}}+e^{u_{t}(t,x)^{2}} \bigr), \quad x \in \mathbf{R}^{2}, t>0 $$utt(t,x)−△u(t,x)=u(t,x)(eu(t,x)2+eut(t,x)2),x∈R2,t>0 with small initial data will blow up in finite time. Moreover, we obtain the lifespan results for the above problems.