Nonexistence of global solutions for an inhomogeneous pseudo-parabolic equation.

In the present paper, we study an inhomogeneous pseudo-parabolic equation with nonlocal nonlinearity u t − k Δ u t − Δ u = I 0 + γ ( | u | p) + ω (x) , (t , x) ∈ (0 , ∞) × R N , where p > 1 , k ≥ 0 , ω (x) ≠ 0 and I 0 + γ is the left Riemann–Liouville fractional integral of order γ ∈ (0 , 1). Based on the test function method, we have proved the blow-up result for the critical case γ = 0 , p = p c for N ≥ 3 , which answers an open question posed by Zhou (2020), and in particular when k = 0 it improves the result obtained by Bandle et al. (2000). An interesting fact is that in the case γ > 0 , the problem does not admit global solutions for any p > 1 and ∫ R N ω (x) d x > 0. [ABSTRACT FROM AUTHOR]