Abstract
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
utt-K(Nu(t))[Delta(p(x))u+Delta(r(x))ut]=F(x,t,u).
Here, K (Nu(t)) is a Kirchhoff function, Delta(r(x))ut 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.