Blow-up analysis for two kinds of nonlinear wave equations

Abstract

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

Keywords

• Wave equation
• Blow up
• Lifespan