## Abstract

We study the parabolic equation u_{t}(t,x)=a^{ij}(t)u_{xixj }(t,x)+f(t,x),(t,x)∈[0,T]×R^{d}u(0,x)=u_{0}(x) with the full degeneracy of the leading coefficients, that is, (a^{ij}(t))≥δ(t)I_{d×d}≥0. It is well known that if f and u_{0} are not smooth enough, say f∈L_{p}(T):=L_{p}([0,T];L_{p}(R^{d})) and u_{0}∈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,⋅)∈L_{p}(R^{d}) on the set {t:δ(t)>0} and ∫0T‖u_{xx}(t)‖_{Lp } ^{p}δ(t)dt≤N(d,p)(∫0T‖f(t)‖_{Lp } ^{p}δ^{1−p}(t)dt+‖u_{0}‖^{p} _{Bp 2−2/p }), where B_{p} ^{2−2/p} is the Besov space of order 2−2/p. We also prove that u_{xx}(t,⋅)∈L_{p}(R^{d}) for all t>0 and ∫0T‖u_{xx}‖_{Lp(Rd)} ^{p}dt≤N‖u_{0}‖_{Bp 2−2/(βp) } ^{p}, if f=0, ∫_{0} ^{t}δ(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 a^{11}(t)=δ(t)=1+sin(1/t), then (0.3) holds with β=1, which actually equals the maximal regularity of the heat equation u_{t}=Δu.

Original language | English |
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Pages (from-to) | 5959-5983 |

Number of pages | 25 |

Journal | Journal of Differential Equations |

Volume | 265 |

Issue number | 11 |

DOIs | |

Publication status | Published - 2018 Dec 5 |

### Bibliographical note

Publisher Copyright:© 2018 Elsevier Inc.

## Keywords

- Initial-value problem
- Maximal L-regularity
- Time degenerate parabolic equations

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics