On the second order derivative estimates for degenerate parabolic equations

Abstract

We study the parabolic equation u(t)(t , x) = a(ij)(t)u(x)i(x)j(t,x) + f (t, x), (t, x)is an element of [0, T] x R-d u(0, x) = u(0)(x) (0,1) with the full degeneracy of the leading coefficients, that is, (a(ij)(t)) >= delta(t)I-dxd >= 0. (0,2) It is well known that if f and up are not smooth enough, say f is an element of L-p(T) := L-p([0, T]; L-p(R-d)) and u(0) is an element of L-p(R-d), then in general the solution is only in C([0, T]; L-p(R-d)), and thus derivative estimates are not possible. In this article we prove that u(xx) (t, center dot) is an element of L-p(R-d) on the set {t : delta (t) > 0} and integral(T)(0)parallel to u(xx)(t)parallel to(p)(Lp) delta(t)dt <= N(d,p) ( integral(T)(0)parallel to f(t)parallel to(p)(Lp) delta 1-p(t)dt + parallel to u(0)parallel to(p)(Bp2-2/p)) where B-p(2-2/p) is the Besov space of order 2 - 2/p. We also prove that u(xx)(t, center dot) is an element of L-p(R-d) for all t > 0 and integral(T)(0)parallel to u(xx)parallel to Lp(Rd)(p)dt <= N parallel to u(0)parallel to(p)(Bp2-2/(beta p)), ((0,3)) if f = 0, integral(t)(0) delta(s)ds > 0 for each t > 0, and a certain asymptotic behavior of delta(t) holds near t = 0 (see (1.3)). Here f > 0 is the constant related to the asymptotic behavior in (1.3). For instance, if d = 1 and a(11)(t) = delta (t) =1 sin(1/t), then (0.3) holds with beta = 1, which actually equals the maximal regularity of the heat equation u(t) = Delta(u). (C) 2018 Elsevier Inc. All rights reserved.