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

TL;DR

This paper investigates blowup phenomena and lifespan estimates for solutions to semilinear wave equations with variable coefficients on exterior domains, establishing critical exponents and demonstrating that solutions cannot exist globally regardless of initial data size.

Contribution

It introduces new blowup results and lifespan estimates for semilinear wave equations with variable coefficients, extending understanding of critical exponents in exterior domains.

Findings

Solutions blow up in finite time for subcritical exponents.

No global solutions exist regardless of initial data size.

Lifespan estimates are provided for solutions.

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 \Omega^{c}\times(0,+\infty), n\geq 3 and u_{tt}-\partial_{i}(a_{ij}(x)\partial_{j}u)=|u_{t}|^{p}, (x,t)\in \Omega^{c}\times (0,+\infty), n\geq 1, where $a_{ij}(x)=\delta_{ij}, when |x|\geq R. The exponents $p$ 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.