Linear parabolic equations with singular potentials. (English) Zbl 1036.35066

Amann, Herbert

J. Evol. Equ. 3, No. 3, 395-406 (2003).

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 [H. Brezis and Th. Cazenave, J. Anal. Math. 68, 277–304 (1996; Zbl 0868.35058); D. Hirata and 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.

Reviewer: Evgeniy A. Kalita (Donetsk)

MSC:

35K20

Initial-boundary value problems for second-order parabolic equations

Keywords:

maximal regularity techniques; solvability; well-posedness

Citations:

Zbl 0868.35058; Zbl 1161.35418