Exponential estimates of solutions of parabolic pseudodifferential equations with discontinuous and growing symbols

Ya Lutsky, V. S. Rabinovich

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Abstract

Let Ω′⊂Rn be an open set, and Ω+ = R+×Ω′ where R+={fx0:x0} We consider pseudodifferential operators in domain Ω+ with double symbols which have singularities near R+×τΩ′ and super exponential growths at infinity. We suppose that symbols have analytic extension with respect to the variable dual to the time in the lower complex half-plane. We construct the theory of invertibility of such operators in weighted Sobolev spaces with weights connected with growths of symbols. We give applications to estimates of the fundamental solutions of such operators, in particular, to the heat equations with singular potentials of power, exponential and super exponential growths.

Original languageEnglish
Pages (from-to)757-778
Number of pages22
JournalComplex Variables and Elliptic Equations
Volume54
Issue number8
DOIs
StatePublished - Aug 2009

Keywords

  • Parabolic differential operators with growing coefficients
  • Parabolic pseudodifferential operators with double symbols
  • Weighted estimates of solutions of the homogeneous Cauchy problem

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