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

Walker, W. J.

doi:10.1090/S0002-9939-1976-0463719-9

Bounds for solutions to ordinary differential equations applied to a singular Cauchy problem
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by W. J. Walker PDF

Proc. Amer. Math. Soc. 54 (1976), 73-79 Request permission

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 \[ y'' + (a(t) + ib(t))y = 0.\]

References

Richard Bellman, Stability theory of differential equations, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953. MR 0061235
Richard Bellman, The boundedness of solutions of linear differential equations, Duke Math. J. 14 (1947), 83–97. MR 21189
I. S. Berezin, On Cauchy’s problem for linear equations of the second order with initial conditions on a parabolic line, Mat. Sbornik N.S. 24(66) (1949), 301–320 (Russian). MR 0031176

Sur l’equation

40

Sopra un criterio di stabilita

11

Robert W. Carroll, Abstract methods in partial differential equations, Harper’s Series in Modern Mathematics, Harper & Row, Publishers, New York-London, 1969. MR 0433480
Robert Carroll, Some degenerate Cauchy problems with operator coefficients, Pacific J. Math. 13 (1963), 471–485. MR 163064
R. W. Carroll and C. L. Wang, On the degenerate Cauchy problem, Canadian J. Math. 17 (1965), 245–256. MR 217399, DOI 10.4153/CJM-1965-023-7
Min’-yu Či, The Cauchy problem for a class of hyperbolic equations with initial data on a line of parabolic degeneracy, Acta Math. Sinica 8 (1958), 521–530 (Chinese, with Russian summary). MR 107088
A. B. Nersesjan, On the Cauchy problem for degenerated hyperbolic equations of second order, Dokl. Akad. Nauk SSSR 166 (1966), 1288–1291 (Russian). MR 0196273
William F. Osgood, On a theorem of oscillation, Bull. Amer. Math. Soc. 25 (1919), no. 5, 216–221. MR 1560178, DOI 10.1090/S0002-9904-1919-03175-9
I. G. Petrovskii, Partial differential equations, Iliffe Books Ltd., London; distributed by W. B. Saunders Co., Philadelphia, Pa., 1967. Translated from the Russian by Scripta Technica, Ltd. MR 0211021
M. H. Protter, The Cauchy problem for a hyperbolic second order equation with data on the parabolic line, Canad. J. Math. 6 (1954), 542–553. MR 64269, DOI 10.4153/cjm-1954-059-x
S. A. Tersenov, A problem with data on a line of degeneration for systems of equations of hyperbolic type, Dokl. Akad. Nauk SSSR 155 (1964), 285–288 (Russian). MR 0164140
W. J. Walker, A stability theorem for a real analytic singular Cauchy problem, Proc. Amer. Math. Soc. 42 (1974), 495–500. MR 342877, DOI 10.1090/S0002-9939-1974-0342877-9
A. Wiman, Über eine Stabilitätsfrage in der Theorie der Linearen Differentialgleichungen, Acta Math. 66 (1936), no. 1, 121–145 (German). MR 1555411, DOI 10.1007/BF02546518