### 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 |
---|---|

Journal | Journal of Differential Equations |

DOIs | |

Publication status | Accepted/In press - 2018 Jan 1 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Analysis

### Cite this

**On the second order derivative estimates for degenerate parabolic equations.** / Kim, Ildoo; Kim, Kyeong Hun.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - On the second order derivative estimates for degenerate parabolic equations

AU - Kim, Ildoo

AU - Kim, Kyeong Hun

PY - 2018/1/1

Y1 - 2018/1/1

N2 - 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.

AB - 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.

KW - Initial-value problem

KW - Maximal L-regularity

KW - Time degenerate parabolic equations

UR - http://www.scopus.com/inward/record.url?scp=85049875876&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85049875876&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2018.07.014

DO - 10.1016/j.jde.2018.07.014

M3 - Article

AN - SCOPUS:85049875876

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

ER -