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


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.


Evolution Equation Parabolic Equation Boundary Problem Compatibility Condition Real Parameter 
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Copyright information

© Nicola Zanichelli Editore 1974

Authors and Affiliations

  • Carlo Pagani
    • 1
  1. 1.Firenze

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