This paper focused on a class of three-dimensional incompressible viscoelastic rate-type fluids with stress diffusion. To simplify the analysis, we considered a model where the elastic stress was spherical. It is worth noting that while the existence of solutions for such fluids has been studied, their stability properties remain largely unexplored. So inspired by ^[1], this paper employed the bootstrap argument, the Schonbek's method, and repeated use of Besov space properties to prove, for the first time, stability of the solution under some additional conditions on the initial data — but without further smallness restrictions. Our results showed that the velocity decays faster than the typical algebraic rate, while the spherical component of the elastic strain tensor exhibited global exponential decay. Finally, using the decay rates, we derived the stability result for any given globally smooth solution—namely, that a sufficiently small perturbation yielded a unique globally smooth solution which stayed close to the original reference solution. We thereby extended the analysis of ^[1] on inhomogeneous Navier-Stokes equations to a viscoelastic fluid with stress diffusion.

We establish the global unique solution for the 2-D inhomogeneous incompressible viscoelastic rate-type fluids with stress-diffusion by employing the standard energy method and the standard compactness arguments.