Kelvin–Voight equation with p-Laplacian and damping term: Existence, uniqueness and blow-up

In this paper, we consider the initial-boundary value problem for a generalized Kelvin–Voight equation with p-Laplacian and a damping term:v→t+(v→⋅∇)v→+∇P(x,t)=ϰdiv(∇v→t)+νdiv(|∇v→|p−2∇v→)+γ|v→|m−2v→,divv→=0. Here v→(x,t) is the velocity field, P(x,t)P(x,t) is the pressure, ν is the viscosity kinematic coefficient, and ϰ is the viscosity relaxation coefficient (is a length scale parameter characterizing the elasticity of the fluid). The coefficient γ and the exponents p, m are given constants. Under appropriate conditions on the data, we prove the existence and uniqueness of the global and local weak solutions. Under several assumptions on the exponents p, m, the coefficients ν, ϰ, and specified initial data, a finite time blow-up and the behavior of the solutions for large times are also established.