Blow-up of solutions to the one-dimensional semilinear wave equation with damping depending on time and space variables

Abstract

In this paper, we give a small data blow-up result for the one-dimensional semilinear wave equation with damping depending on time and space variables. We show that if the damping term can be regarded as perturbation, that is, non-effective damping in a certain sense, then the solution blows up in finite time for any power of nonlinearity. This gives an affirmative answer for the conjecture that the critical exponent agrees with that of the wave equation when the damping is non-effective in one space dimension.

15 pages. We correct the definition of test functions to treat the critical case in the proof of Theorems 1.5 and 1.7

References in corpus (4)

• The Threshold between Effective and Noneffective Damping for Semilinear Waves
• Semi-linear wave equations with effective damping
• A note on the lifespan of solutions to the semilinear damped wave equation
• Small data global existence for the semilinear wave equation with space-time dependent damping

Cited by in corpus (2)

• Scaling variables and asymptotic profiles for the semilinear damped wave equation with variable coefficients
• Critical exponent for the semilinear wave equations with a damping increasing in the far field