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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,+∞), $$\begin{aligned} &u_{t}-\Delta u + \int _{0}^{t} {{g_{1}}(t - s) \operatorname{div}\bigl({a_{1}}(x) \nabla u(s)\bigr)\,ds} \\ &\quad{} + \int _{0}^{ + \infty } {{g_{2}}(s) \operatorname{div}\bigl({a_{2}}(x)\nabla u(t - s)\bigr)\,ds}+ f(u)=0, \quad(x,t)\in \varOmega \times [0,+\infty ), \end{aligned}$$ where Ω is a bounded domain. Under suitable conditions on a1$a_{1}$ and a2$a_{2}$, for a large class of relation functions g1$g_{1}$ and g2$g_{2}$, we establish a general decay estimate, including the usual exponential and polynomial decay cases.

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