Existence and general decay for nondissipative distributed systems with boundary frictional and memory dampings and acoustic boundary conditions

In this paper, we study the existence and general energy decay rate of global solutions for nondissipative distributed systems u′′-▵u+h(∇u)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u''-\triangle u+h(\nabla u)=0$$\end{document}with boundary frictional and memory dampings and acoustic boundary conditions. For the existence of solutions, we prove the global existence of weak solution by using Faedo–Galerkin’s method and compactness arguments. For the energy decay rate, we first consider the general nonlinear case of h satisfying a smallness condition and prove the general energy decay rate by using perturbed modified energy method. Then, we consider the linear case of h: h(∇u)=-∇ϕ·∇u\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${h(\nabla u)=-\nabla\phi\cdot\nabla u}$$\end{document} and prove the general decay estimates of equivalent energy.