(untitled) (Journal of Partial Differential Equations, 2021)
In this paper, we consider a class of Kirchhoff equation, in the presence of a Kelvin-Voigt type damping and a source term of general nonlinearity forms. Where the studied equation is given as follows equation*u_tt -K( Nu(t))[ Δ_p(x)u +Δ_r(x)u_t]=F(x, t, u).equation* Here, K( Nu(t)) is a Kirchhoff function, Δ_r(x)u_t represent a Kelvin-Voigt strong damping term, and F(x, t, u) is a source term. According to an appropriate assumption, we obtain the local existence of the weak solutions by applying the Galerkin's approximation method. Furthermore, we prove a non-global existence result for certain solutions with negative/positive initial energy. More precisely, our aim is to find a sufficient conditions for p(x), q(x), r(x), F(x,t,u) and the initial data for which the blow-up occurs.