On the second order derivative estimates for degenerate parabolic equations

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Abstract

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+‖u0p 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‖uxxLp(Rd) pdt≤N‖u0Bp 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.

Original languageEnglish
JournalJournal of Differential Equations
DOIs
Publication statusAccepted/In press - 2018 Jan 1

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Degenerate Parabolic Equation
Second-order Derivatives
Asymptotic Behavior
Coversine
Maximal Regularity
Besov Spaces
Degeneracy
Heat Equation
Estimate
Parabolic Equation
Derivative
Coefficient

Keywords

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

ASJC Scopus subject areas

  • Analysis

Cite this

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title = "On the second order derivative estimates for degenerate parabolic equations",
abstract = "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.",
keywords = "Initial-value problem, Maximal L-regularity, Time degenerate parabolic equations",
author = "Ildoo Kim and Kim, {Kyeong Hun}",
year = "2018",
month = "1",
day = "1",
doi = "10.1016/j.jde.2018.07.014",
language = "English",
journal = "Journal of Differential Equations",
issn = "0022-0396",
publisher = "Academic Press Inc.",

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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

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