EXPONENTIAL STABILITY FOR THE GENERALIZED KIRCHHOFF TYPE EQUATION IN THE PRESENCE OF PAST AND FINITE HISTORY

In this paper, we study the generalized Kirchhoff type equation in the presence of past and finite history $$\large u_{tt}-M(x,t,{\tau},\;{\parallel}{\nabla}u(t){\parallel}^2){\Delta}u+{\normalsize\displaystyle\smashmargin{2}{\int\nolimits_0}^t}\;h(t-{\tau})div[a(x){\nabla}u({\tau})]d{\tau}\\\hspace{25}-{\normalsize\displaystyle\smashmargin{2}{\int\nolimits_{-{\infty}}}^t}\;k(t-{\tau}){\Delta}u(x,t)d{\tau}+{\mid}u{\mid}^{\gamma}u+{\mu}_1u_t(x,t)+{\mu}_2u_t(x,t-s(t))=0.$$ Under the smallness condition with respect to Kirchhoff coefficient and the relaxation function and other assumptions, we prove the expoential decay rate of the Kirchhoff type energy.