On the second order derivative estimates for degenerate parabolic equations

In this article we prove that uxx(t,⋅)∈Lp(Rd) on the set {t:δ(t)>0} and∫0T‖uxx(t)‖Lppδ(t)dt≤N(d,p)(∫0T‖f(t)‖Lppδ1−p(t)dt+‖u0‖pBp2−2/p), where Bp2−2/p is the Besov space of order 2−2/p. We also prove that uxx(t,⋅)∈Lp(Rd) for all t>0 and(0.3)∫0T‖uxx‖Lp(Rd)pdt≤N‖u0‖Bp2−2/(βp)p, if f=0, ∫0tδ(s)ds>0 for each t>0, and a certain asymptotic behavior of δ(t) holds near t=0 (see (1.3)). Here β>0 is the constant related to the asymptotic behavior in (1.3). For instance, if d=1 and a11(t)=δ(t)=1+sin⁡(1/t), then (0.3) holds with β=1, which actually equals the maximal regularity of the heat equation ut=Δu.