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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: The Cauchy problem $ {u_{tt}} - {t^{2 + \varepsilon }}{u_{xx}} - {u_y} = 0,\varepsilon > 0,u(x,y,0) = \alpha (x,y),{u_t}(x,y,0) = \gamma (x,y)$, is shown to be unstable by demonstrating that there exists a sequence of solutions which increase indefinitely on a sequence of neighbourhoods of $ t = 0$ which shrink to zero, while at the same time the initial data is tending to zero. The equation $ {u_{tt}} - {t^{2 + \varepsilon }}{u_{xx}} - {u_{yy}} - {u_y} = 0$ 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 $ t = 0$ 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 $ t = 0$ for complex-valued solutions of the ordinary differential equation

$\displaystyle y'' + (a(t) + ib(t))y = 0.$


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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

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