Abstract: We investigate the blow-up effect of solutions for a non-homogeneous wave equation
u_tt −∆u−∆u_t =I₀^α+ (|u|^p)+ω(x),
where p >1, 0≤α<1 and ω(x) with \int_{\mathbb{R}^{N}} ω(x)dx >0. By a way of combining the argument by contradiction with the test function techniques, we prove that not only any non-trivial solution blows up in finite time under 0< α <1, N ≥1 and p >1, but also any non-trivial solution blows up in finite time under α= 0, 2≤ N ≤4 and p being the Strauss exponent.

Key words: Finite time blow-up, Non-homogeneously strongly damped wave equation; Riemann-Liouville fractional integral, Strauss exponent