In this talk, I'll present our recent work on the lifespan estimates for power type nonlinear wave equations with damping and potential terms. The damping and potential functions are supposed to be time-independent and have certain decay at spatial infinity. We will propose some "critical" powers which relates with the decay rate. The key ingredient is the analysis of the asymptotic behavior for special solutions to a class of elliptic "eigenvalue" problems, which will be used to construct test functions. It is based on my recent work with Ning-An Lai, Mengyun Liu and Ziheng Tu. 

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