Linear parabolic equations with singular potentials. (Q1412087)

The initial boundary value problem for the equation \[ u_t -\Delta u +a(x,t)u=0 \] is considered. Minimal regularity of the `potential' \(a\) and initial boundary data providing well-posedness of the problem are discussed. Existence and uniqueness results for \(L_r(L_q)\)-solution are established. In contrast to previous works [\textit{H. Brezis} and \textit{Th. Cazenave}, J. Anal. Math. 68, 277--304 (1996; Zbl 0868.35058); \textit{D. Hirata} and \textit{M. Tsutsumi}, Differ. Integral Equ. 14, 1--18 (2001; Zbl 1161.35418)] which rely on a priori estimates and properties of the heat semigroup, this paper employs maximal regularity techniques. It allows to get far reaching generalizations and improvements of previous results.