Blow‐Up Criteria for Higher Order Fractional Moore–Gibson–Thompson Equations With Fractional Laplacian Damping

This paper presents the first systematic analysis of critical exponents for both third and fourth‐order fractional Moore–Gibson–Thompson (MGT) equations with combined spatiotemporal fractional derivatives and structural damping. We establish sharp nonexistence results for global solutions via an adapted test function method, deriving explicit critical exponent formulas that quantify the interplay between fractional orders ( α , β , γ , δ ), spatial dimension N , and the nonlinearity exponent k . Notably, our framework accommodates the nonmonotone source term | ψ | k |1 − ψ | ϑ , extending prior results for integer‐order MGT and fractional wave equations.