Decay rates to equilibrium in a nonlinear subdiffusion equation with two counteracting terms

In this paper we prove convergence to a steady state as $t\to \mathrm{\infty }$ for solutions to the subdiffusion equation $${\mathrm{\partial }}_t^{\alpha }u-\mathbb{L}u=q(x)u-p(x)f(u)+r$$ with the exponential ($\alpha =1$) or power law ($\alpha \in [0,1)$) rates under mild conditions on the coefficients $p$, $q$, the nonlinearity $f$, the source $r$, and the elliptic operator $\mathbb{L}$.