Blow-up of solutions for coupled wave equations with damping terms and derivative nonlinearities

Abstract

This work was concerned with the weakly coupled system of semi-linear wave equations with time dependent speeds of propagation, damping terms, and derivative nonlinear terms in generalized Einstein-de Sitter space-time on $ \mathbb{R}^n $. Under certain assumptions about the indexes $ k_1, \, k_2 $, coefficients $ \mu_1, \, \mu_2 $, and nonlinearity exponents $ p, \, q $, applying the iteration technique, finite time blow-up of local solutions to the small initial value problem of the coupled system was investigated. Blow-up region and upper bound lifespan estimate of solutions to the problem were established. Compared with blow-up results in the previous literature, the new ingredient relied on that the blow-up region of solutions obtained in this work varies due to the influence of coefficients $ k_1, \, k_2 $.

Keywords

• coupled wave equations
• damping terms
• iteration method
• blow-up
• lifespan estimate