On the second order derivative estimates for degenerate parabolic equations

We study the parabolic equation ut(t,x)=aij(t)uxixj (t,x)+f(t,x),(t,x)∈[0,T]×Rdu(0,x)=u0(x) with the full degeneracy of the leading coefficients, that is, (aij(t))≥δ(t)Id×d≥0. It is well known that if f and u0 are not smooth enough, say f∈Lp(T):=Lp([0,T];Lp(Rd)) and u0∈Lp(Rd), then in general the solution is only in C([0,T];Lp(Rd)), and thus derivative estimates are not possible. In this article we prove that uxx(t,⋅)∈Lp(Rd) on the set {t:δ(t)>0} and ∫0T‖uxx(t)‖Lp pδ(t)dt≤N(d,p)(∫0T‖f(t)‖Lp pδ1−p(t)dt+‖u0‖p Bp 2−2/p ), where Bp 2−2/p is the Besov space of order 2−2/p. We also prove that uxx(t,⋅)∈Lp(Rd) for all t>0 and ∫0T‖uxx‖Lp(Rd) pdt≤N‖u0‖Bp 2−2/(βp) p, if f=0, ∫0 tδ(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.