Suppose \(u = u(t;u_0)\) solves the heat equation \(u_t - \Delta u = 0\) with initial condition \(u_0 \ge 0\) and Dirichlet boundary conditions in a bounded, sufficiently smooth domain (say \(C^2\)) in \(\mathbb R^N\). Then for each \(t>0\) there exist constants \(0 < c_t < C_t\) such that \(c_t \varphi \le u(t) \leq C_t \varphi \), where \(\varphi \) denotes the principal eigenfunction of the Dirichlet Laplacian (a.k.a the “ground state”). Such a result is well-known in the study of diffusion equations (see e.g. Theorem 4.2.5 in the classic book on heat kernels by Davies). Typically, the upper bound is established first, and then there are various methods to obtain the lower bound based on spectral theory and properties of the heat kernel.
In this talk, we present a functional analytic approach to this phenomenon of lower bounds obtained from upper bounds. Our proof relies solely on properties of positivity-preserving operators, and thus shows that the phenomenon occurs in far greater generality than is considered in the classical case.
This is ongoing work with Daniel Daners and Jochen Glück.