Decay rates to equilibrium in a nonlinear subdiffusion equation with two counteracting terms
Barbara Kaltenbacher
TL;DR
The paper studies long-time behavior of solutions to a nonlinear time-fractional subdiffusion equation with spatially varying counteracting terms, proving convergence to a steady state $u^\infty$ as $t\to\infty$. By developing energy estimates that exploit coercivity of the elliptic operator $\mathbb{L}$, a barrier argument for nonlinearities, and Tauberian tools, it establishes decay rates for the error $u-u^\infty$ in Sobolev norms, with $\|u(t)-u^\infty\|_{H^s(\Omega)}^2 \le C\Psi(t)$ where $\Psi(t)=t^{-\alpha}$ for $\alpha\in(0,1)$ and $\Psi(t)=e^{-\omega t}$ for $\alpha=1$. The results allow low-regularity coefficients $p$ and $q$ (e.g., $p\in L^{\mathfrak{r}}(\Omega)$) and accommodate sources $r$ approaching $r^\infty$, under a sign condition on the linearization around $u^\infty$. A no-smallness variant extends decay results without assuming small initial perturbations, and decay of time derivatives is also characterized, making the findings relevant for inverse problems and fixed-point reconstructions where convergence to equilibrium is essential.
Abstract
In this paper we prove convergence to a steady state as $t\to\infty$ for solutions to the subdiffusion equation \[ \partial_t^αu - \mathbb{L} u = q(x)u - p(x)f(u) + r \] with the exponential ($α=1$) or power law ($α\in[0,1)$) rates under mild conditions on the coefficients $p$, $q$, the nonlinearity $f$, the source $r$, and the elliptic operator $\mathbb{L}$.