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Annali di Matematica Pura ed Applicata

, Volume 99, Issue 1, pp 333–399 | Cite as

On the parabolic equation sgn (x)|x|p u y u xx =0 and a related one

  • Carlo Pagani
Article

Summary

Boundary value problems in the half-plane y>0 for the evolution equation: |x|puy−uxx=f and for the forward-backward equation:sgn(x)|x|puy−uxx=0 (where p is a real parameter >−1 and f is a given function) are investigated. We prove the uniquenes of solutions u, whose generalized derivatives uy and uxx are square integrable with suitable weights. We prove also the existence of such a solution for the evolution equation and we show that the boundary problem for the forward-backward equation has an index different from zero; the compatibility conditions which must be imposed on the boundary data in order that a solution exists with a prescribed regularity are explicitly given. To prove these results, we make use substantially of an integral transformation technique; thus some special integral transformations of L2(0,+∞) onto (or into) itself are studied. Moreover a boundary problem for the half-plane x>−L (L>0) is also discussed.

Keywords

Evolution Equation Parabolic Equation Boundary Problem Compatibility Condition Real Parameter 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Nicola Zanichelli Editore 1974

Authors and Affiliations

  • Carlo Pagani
    • 1
  1. 1.Firenze

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