### Abstract

We construct algebras of Volterra pseudodifferential operators that contain, in particular, the inverses of the most natural classical systems of parabolic boundary value problems of general form. Parabolicity is determined by the invertibility of the principal symbols, and as a result, is equivalent to the invertibility of the operators within the calculus. Existence, uniqueness, regularity, and asymptotics of solutions as t → ∞ are consequences of the mapping properties of the operators in exponentially weighted Sobolev spaces and subspaces with asymptotics. An important aspect of this work is that the microlocal and global kernel structure of the inverse operator (solution operator) of a parabolic boundary value problem for large times is clarified. Moreover, our approach naturally yields qualitative perturbation results for the solvability theory of parabolic boundary value problems. To achieve these results, we assign to t = ∞ the meaning of a conical point and treat the operators as totally characteristic pseudodifferential boundary value problems.

Original language | English (US) |
---|---|

Pages (from-to) | 91-163 |

Number of pages | 73 |

Journal | Japanese Journal of Mathematics |

Volume | 30 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2004 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

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*Japanese Journal of Mathematics*, vol. 30, no. 1, pp. 91-163. https://doi.org/10.4099/math1924.30.91

**On the inverse of parabolic boundary value problems for large times.** / Krainer, Thomas.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On the inverse of parabolic boundary value problems for large times

AU - Krainer, Thomas

PY - 2004/1/1

Y1 - 2004/1/1

N2 - We construct algebras of Volterra pseudodifferential operators that contain, in particular, the inverses of the most natural classical systems of parabolic boundary value problems of general form. Parabolicity is determined by the invertibility of the principal symbols, and as a result, is equivalent to the invertibility of the operators within the calculus. Existence, uniqueness, regularity, and asymptotics of solutions as t → ∞ are consequences of the mapping properties of the operators in exponentially weighted Sobolev spaces and subspaces with asymptotics. An important aspect of this work is that the microlocal and global kernel structure of the inverse operator (solution operator) of a parabolic boundary value problem for large times is clarified. Moreover, our approach naturally yields qualitative perturbation results for the solvability theory of parabolic boundary value problems. To achieve these results, we assign to t = ∞ the meaning of a conical point and treat the operators as totally characteristic pseudodifferential boundary value problems.

AB - We construct algebras of Volterra pseudodifferential operators that contain, in particular, the inverses of the most natural classical systems of parabolic boundary value problems of general form. Parabolicity is determined by the invertibility of the principal symbols, and as a result, is equivalent to the invertibility of the operators within the calculus. Existence, uniqueness, regularity, and asymptotics of solutions as t → ∞ are consequences of the mapping properties of the operators in exponentially weighted Sobolev spaces and subspaces with asymptotics. An important aspect of this work is that the microlocal and global kernel structure of the inverse operator (solution operator) of a parabolic boundary value problem for large times is clarified. Moreover, our approach naturally yields qualitative perturbation results for the solvability theory of parabolic boundary value problems. To achieve these results, we assign to t = ∞ the meaning of a conical point and treat the operators as totally characteristic pseudodifferential boundary value problems.

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U2 - 10.4099/math1924.30.91

DO - 10.4099/math1924.30.91

M3 - Article

VL - 30

SP - 91

EP - 163

JO - Japanese Journal of Mathematics

JF - Japanese Journal of Mathematics

SN - 0289-2316

IS - 1

ER -