Global existence and blow-up of solutions for mixed local and nonlocal hyperbolic equations

In this paper, we consider the following mixed local and nonlocal hyperbolic equation: u t t − Δ u + μ ( − Δ ) s u = | u | p − 2 u , in Ω × R + , u ( x , 0 ) = u 0 ( x ) , u t ( x , 0 ) = u 1 ( x ) , in Ω , u ( x , t ) = 0 , in ( R N ∖ Ω ) × R 0 + , where s ∈ ( 0 , 1 ), N > 2, p ∈ ( 2 , 2 s ∗ ], μ is a nonnegative real parameter, Ω ⊂ R N is a bounded domain with Lipschitz boundary ∂ Ω, Δ is the Laplace operator, ( − Δ ) s is the fractional Laplace operator. By combining the Galerkin approach with the modified potential well method, we obtain the global existence, vacuum isolating, and blow-up of solutions for the aforementioned problem, provided certain assumptions are fulfilled. Specifically, we study the existence of global solutions for the above problem in the cases of subcritical and critical initial energy levels, as well as the finite time blow-up of solutions. Then, we investigate the blow-up of solutions for the above problem in the case of supercritical initial energy level, as well as upper and lower bounds of blow-up time of solutions.