AbstractWe prove global solvability of the third-order in time Jordan–More–Gibson–Thompson acoustic wave equation with memory in $${\mathbb {R}}^n$$ R n , where $$n \ge 3$$ n ≥ 3 . This wave equation models ultrasonic propagation in relaxing hereditary fluids and incorporates both local and cumulative nonlinear effects. The proof of global existence is based on a sequence of high-order energy bounds that are uniform in time, and derived under the assumption of an exponentially decaying memory kernel and sufficiently small and regular initial data.