Bounds for solutions to ordinary differential equations applied to a singular Cauchy problem

Author:
W. J. Walker

Journal:
Proc. Amer. Math. Soc. **54** (1976), 73-79

MSC:
Primary 35M05

DOI:
https://doi.org/10.1090/S0002-9939-1976-0463719-9

MathSciNet review:
0463719

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Abstract | References | Similar Articles | Additional Information

Abstract: The Cauchy problem , is shown to be unstable by demonstrating that there exists a sequence of solutions which increase indefinitely on a sequence of neighbourhoods of which shrink to zero, while at the same time the initial data is tending to zero. The equation is investigated with the same initial data and in this case it is shown that the sequence of solutions remains bounded on a neighbourhood of which suggests but does *not* prove that the Cauchy problem for this equation is well posed. The latter result is a consequence of bounds obtained on a neighbourhood of for complex-valued solutions of the ordinary differential equation

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1976-0463719-9

Keywords:
Cauchy problem,
parabolic degeneracy,
three independent variables,
dependence on initial conditions

Article copyright:
© Copyright 1976
American Mathematical Society