Lifespan of Classical Solutions to Semilinear Neumann-Wave Equations

We study the initial-boundary value problems of quadratic semilinear wave equations outside of nontrapping obstacles with Neumann boundary conditions in spatial dimensions n >= 3. We obtain some lower bounds of the lifespan of classical solutions, which coincide with the sharp results for the corresponding Cauchy problems. Particularly, our results generalize the works of Shibata and Tsutsumi (North-Holland Mathematics Studies, vol 128, pp 175-228, 1985) in n >= 6, of Hayashi (J. Funct. Anal. 131:302-344, 1995) in n = 4, 5, and of Godin (Comm Partial Differ Equ 14:299-374, 1989) in n = 3.