Blow up of Solutions to Semilinear Wave Equations with variable coefficients and boundary

Abstract

This paper is devoted to studying the following two initial-boundary value problems for semilinear wave equations with variable coefficients on exterior domain with subcritical exponent in $n$ space dimensions: u_{tt}-partial_{i}(a_{ij}(x)\partial_{j}u)=|u|^{p}, (x,t)\in Ω^{c}\times(0,+\infty), n\geq 3 and u_{tt}-\partial_{i}(a_{ij}(x)\partial_{j}u)=|u_{t}|^{p}, (x,t)\in Ω^{c}\times (0,+\infty), n\geq 1, where $a_{ij}(x)={\delta }_{ij},when|x|\ge R.Theexponents$p$satisfies$ 1<p<p_{1}(n)$in(0.1),and$p \leq p_{2}(n)$in(0.2),where$p_{1}(n)$ is the larger root of the quadratic equation (n-1)p^{2}-(n+1)p-2=0, and p_{2}(n)=\frac{2}{n-1}+1, respectively. It is well-known that the numbers p_{1}(n) and p_{2}(n) are the critical exponents. We will establish two blowup results for the above two initial-boundary value problems, it is proved that there can be no global solutions no matter how small the initial data are, and also we give the lifespan estimate of solutions for above problems.