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

In this paper we prove convergence to a steady state as $t\to\infty$ for solutions to the subdiffusion equation $$\partial_t^\alpha u - \mathbb{L} u = q(x)u - p(x)f(u) + r$$ 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 $\mathbb{L}$.