Global Solution of a Non-Linear Kirchhoff – Carrier Type Equation with Dissipation

This paper demonstrates the existence and uniqueness of a local weak solution for the mixed problem (P). This problem is described in Equation (1). To assess the existence of global solutions, we used the Faedo-Galerkin method, the Aubin-Lions theorem of compactness, and important inequalities of functional analysis. Additionally, with respect to the uniqueness of the global solutions, we used the energy method due to the solution's regularity. We also used some inequalities of functional analysis. We use Nakao's lemma and some functional analysis inequalities to treat the asymptotic behavior of the problem.