This paper investigates the Cauchy problem for the semilinear damped wave equation uₓₓ+L₀, ₁u+uₜ=|u|ᵖ with the mixed local-nonlocal operator L₀, ₁: =-aΔ+b (-Δ) ^σ, where a, b_+ and σ (0, 1) (1, +). We determine the critical exponent for this problem being p₂ₑ₈ₓ=1+2\1, σ\n, which sharply separates global in-time existence and finite-time blow-up of solutions. Furthermore, for the super-critical case p>p₂ₑ₈ₓ, we establish the asymptotic profiles of global in-time solutions, showing the anomalous diffusion when σ (0, 1) and the classical diffusion when σ (1, +), together with the sharp decay estimates. For solutions blowing up in finite time when 1<p p₂ₑ₈ₓ, we derive the sharp estimates for upper and lower bounds of lifespan. Our results reveal the crucial influence of mixed operators on the qualitative properties of solutions, fundamentally governing their critical phenomena, large-time behavior and blow-up dynamics, via \1, σ\ or \1, σ\.
Chen et al. (Mon,) studied this question.