The iterated abstract euler-poisson-darboux problem for negative values of the parameter

Abstract

Let X be a Banach space and let A=B² in which B is the infinitesimal generator of a strongly continous group in X for α>0.We construct solution representations of the iterated Cauchy problem

in which the π_j∊D(A^r) for r ≥n.When α=2m+1.these initial conditions can be satisfied only if n-1≤m. This corresponds to the exceptional Euler poisson-Darboux problem. The representations are obtained by using the models of associated equations and parametric differentation and can be applied to the treatment of the iterated GASPT equation and associated function theory.

Keywords:

• Euler-Poisson-Darboux problem
• iterated equation
• exceptional problem
• associated equation
• parametric differentiation
• logarithmic singularity
• GASPT equation

AMS(MOS):

• 35Q05
• 35L30
• 35L67