Exponential stability, change of stability and eigenvalue problems for linear time-periodic parabolic equations on \(\mathbb{R}^ N\) (Q1328960)

The main aim of the authors is to give necessary and sufficient conditions for the nonnegative weight function \(m(x,t)\) in order that the zero solution of the linear time-periodic equation \(\partial_ t u- \Delta u=- m(x,t)u\), \((x,t)\in \mathbb{R}^ N\times (0,\infty)\) be exponentially stable. The result is applied to the study of the stability properties of the parameter-dependent linear time-periodic equation \(\partial_ t u-\Delta u= \lambda m(x,t)u\), \((x,t)\in \mathbb{R}^ N\times (0,\infty)\), where \(\lambda>0\). They suppose that \(m(x,t)\) belongs to the space of \(\nu\)-Hölder continuous and \(T\)-periodic functions taking values in the space of bounded uniformly continuous functions on \(\mathbb{R}^ N\).