A general decay result for a semilinear heat equation with past and finite history memories

In this paper, we consider the initial-boundary value problem of the following semilinear heat equation with past and finite history memories: ut−Δu+∫0tg1(t−s)div(a1(x)∇u(s))ds+∫0+∞g2(s)div(a2(x)∇u(t−s))ds+f(u)=0,(x,t)∈Ω×[0,+∞),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} &u_{t}-\Delta u + \int _{0}^{t} {{g_{1}}(t - s) \operatorname{div}\bigl({a_{1}}(x) \nabla u(s)\bigr)\,ds} \\ &\quad{} + \int _{0}^{ + \infty } {{g_{2}}(s) \operatorname{div}\bigl({a_{2}}(x)\nabla u(t - s)\bigr)\,ds}+ f(u)=0, \quad(x,t)\in \varOmega \times [0,+\infty ), \end{aligned}$$ \end{document} where Ω is a bounded domain. Under suitable conditions on a1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a_{1}$\end{document} and a2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a_{2}$\end{document}, for a large class of relation functions g1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$g_{1}$\end{document} and g2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$g_{2}$\end{document}, we establish a general decay estimate, including the usual exponential and polynomial decay cases.