Absorption evolution families and exponential stability of non-autonomous diffusion equations

Abstract

For a positive, strongly continuous evolution family $\mathcal{U}$ on $L^p(\Omega)$ and a positive, measurable, time dependent potential $V(\cdot)$ we construct a corresponding absorption evolution family $\mathcal {U}_V$ by a procedure introduced by J. Voigt, [24, 25], in the autonomous case. We give sufficient conditions on $\mathcal{U}$ and $V$ such that $\mathcal {U}_V$ is strongly continuous and satisfies variation of constants formulas. For an evolution family $\mathcal {U}$ on $L^1(\mathbb{R}^N)$ satisfying upper and lower Gaussian estimates exponential stability of $\mathcal {U}_V$ is characterized by a condition on the size of $V$ extending recent results by W.Arendt and C.J.K. Batty, [2, 3, 6], in the autonomous case and by D. Daners, M. Hieber, P. Koch Medina, and S. Merino, [8, 12], in the time periodic case. An application to a second order parabolic equation with real coefficients and singular potential is given.