Existence of solutions to heat equations with singular lower order terms

This paper is concerned with the solvability for the Cauchy problem of heat equations in RNRN (N⩾2N⩾2) with singular lower order terms:{∂u∂t(x,t)=Δu(x,t)−βx|x|2⋅∇u(x,t)+c|x|2u(x,t)+f(x,t)in RN×(0,T),u(x,0)=u0(x)in RN, where β∈Rβ∈R, c⩽(N−2−β)2/4c⩽(N−2−β)2/4, u0u0 is an initial value, f   is an inhomogeneous term and T∈(0,∞]T∈(0,∞] is arbitrarily given. In particular, u0u0 and f   are taken from a weighted LpLp-space. The result generalizes the one of Baras and Goldstein (1984) [2] from β=0β=0 to β≠0β≠0 and improves the one of Arendt, Goldstein and Goldstein (2006) [1] which deals with an m  -accretive realization of the linear part in a weighted LpLp-space under the stronger condition c⩽(N−2−β)2(p−1)/p2c⩽(N−2−β)2(p−1)/p2. Moreover, this paper clarifies some regularities of solutions.