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On a singular parabolic equation related to axially symmetric heat potentials

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Summary

An initial value problem for the following singular parabolic differential equation

$$u_{x\alpha } + \frac{{2v}}{x}u_x = u_t (2v > - 1)$$

is studied in some suitable weighted Sobolev class. Such an equation is related to axially symmetric problems and also occurs in probability theory. Uniqueness, existence and representation theorems for the solution are proved and differentiability and regularity properties are investigated. The non-homogeneous equation is also considered.

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Entrata in Redazione il 24 aprile 1974.

Lavoro eseguito con il contributo del C.N.R. nell’ambito dell’attività del Gruppo Nazionale per l’Analisi Funzionale e le sue Applicazioni.

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Arena, O. On a singular parabolic equation related to axially symmetric heat potentials. Annali di Matematica 105, 347–393 (1975). https://doi.org/10.1007/BF02414938

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Keywords

  • Differential Equation
  • Probability Theory
  • Parabolic Equation
  • Representation Theorem
  • Regularity Property