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

Full-text PDF Free Access

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

**[1]**R. Bellman,*Stability theory of differential equations*, McGraw-Hill, New York, 1953. MR**15**, 794. MR**0061235 (15:794b)****[2]**-,*The boundedness of solutions of linear differential equations*, Duke Math. J.**14**(1947), 83-97. MR**9**, 35. MR**0021189 (9:35b)****[3]**I. S. Berezin,*On Cauchy's problem for linear equations of the second order with initial conditions on a parabolic line*, Mat. Sb.**24**(**66**)(1949), 301-320; English transl., Amer. Math. Soc. Transl. (1)**4**(1962), 415-439. MR**11**, 112. MR**0031176 (11:112c)****[4]**M. Biernacki,*Sur l'equation*, Prace Mat. Fiz.**40**(1932), 163-171.**[5]**R. Cacciopoli,*Sopra un criterio di stabilita*, Rend. Accad. Lincei Roma**11**(1930), 251-254.**[6]**R. W. Carroll,*Abstract methods in partial differential equations*, Harper & Row, New York, 1969. MR**0433480 (55:6456)****[7]**-,*Some degenerate Cauchy problems with operator coefficients*, Pacific J. Math.**13**(1963), 471-485. MR**29**#367. MR**0163064 (29:367)****[8]**R. W. Carroll and C. L. Wang,*On the degenerate Cauchy problem*, Canad. J. Math.**17**(1965), 245-256. MR**36**#489. MR**0217399 (36:489)****[9]**Chi Min-you,*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 Math. Acta**9**(1967), 246-254. MR**21**#5815. MR**0107088 (21:5815)****[10]**A. B. Nersesjan,*The Cauchy problem for degenerating hyperbolic equations of second order*, Dokl. Akad. Nauk SSSR**166**(1966), 1288-1291 = Soviet Math. Dokl.**7**(1966), 278-281. MR**33**#4465. MR**0196273 (33:4465)****[11]**W. F. Osgood,*On a theorem of oscillation*, Bull. Amer. Math. Soc.**25**(1919), 216-221. MR**1560178****[12]**I. G. Petrovskii,*Partial differential equations*, 3rd ed., Fizmatgiz, Moscow, 1961; English transl., Scripta Technica; distributed by Saunders, Philadelphia, Pa., 1967. MR**25**#2308;**35**#1906. MR**0211021 (35:1906)****[13]**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**16**, 255. MR**0064269 (16:255d)****[14]**S. A. Tersenov,*A problem with data given on a line of degeneracy for a system of hyperbolic equations*, Dokl. Akad. Nauk SSSR**155**(1964), 285-288 = Soviet Math. Dokl.**5**(1964), 409-412. MR**29**# 1439. MR**0164140 (29:1439)****[15]**W. J. Walker,*A stability theorem for a real analytic singular Cauchy problem*, Proc. Amer. Math. Soc.**42**(1974), 495-500. MR**0342877 (49:7621)****[16]**A. Wiman,*Über eine Stabilitätsfrage in der theorie der linearen Differentialgleichungen*, Acta Math.**66**(1936), 121-145. MR**1555411**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
35M05

Retrieve articles in all journals with MSC: 35M05

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