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

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.

### 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
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