Some remarks on uniqueness for a class of singular abstract Cauchy problems
HTML articles powered by AMS MathViewer
- by James A. Donaldson and Jerome A. Goldstein PDF
- Proc. Amer. Math. Soc. 54 (1976), 149-153 Request permission
Abstract:
Of concern is the Cauchy problem for equations of the form $u''(t) + \alpha (t)u’(t) + {S^2}u(t) = 0(’ = d/dt)$ on a complex Hilbert space $X$. $S$ is a selfadjoint operator on $X$ while $\alpha$ is a continuous function on $(0,\infty )$ which can be unbounded at $t = 0$. Uniqueness results are obtained for these equations by applying a uniqueness theorem for nonlinear equations. Furthermore, nonuniqueness examples for the linear abstract Euler-Poisson-Darboux equation, which is contained in this class, show that the uniqueness theorem is best possible.References
- E. K. Blum, The Euler-Poisson-Darboux equation in the exceptional cases, Proc. Amer. Math. Soc. 5 (1954), 511–520. MR 63543, DOI 10.1090/S0002-9939-1954-0063543-0
- Robert Carroll, On the singular Cauchy problem, J. Math. Mech. 12 (1963), 69–102. MR 0147770
- Robert W. Carroll, Some singular Cauchy problems, Ann. Mat. Pura Appl. (4) 56 (1961), 1–31. MR 151722, DOI 10.1007/BF02414262
- Robert Carroll, Quelques problémes de Cauchy singulariers, C. R. Acad. Sci. Paris 251 (1960), 498–500. MR 113036
- Robert Carroll, Some singular mixed problems, Proc. Nat. Acad. Sci. U.S.A. 46 (1960), 1594–1596. MR 131674, DOI 10.1073/pnas.46.12.1594
- Solange Delache and Jean Leray, Calcul de la solution élémentaire de l’opérateur d’Euler-Poisson-Darboux et de l’opérateur de Tricomi-Clairaut, hyperbolique, d’ordre $2$, Bull. Soc. Math. France 99 (1971), 313–336 (French). MR 303123
- J. B. Diaz and H. F. Weinberger, A solution of the singular initial value problem for the Euler-Poisson-Darboux equation, Proc. Amer. Math. Soc. 4 (1953), 703–715. MR 58099, DOI 10.1090/S0002-9939-1953-0058099-1
- J. A. Donaldson, A singular abstract Cauchy problem, Proc. Nat. Acad. Sci. U.S.A. 66 (1970), 269–274. MR 265797, DOI 10.1073/pnas.66.2.269
- J. A. Donaldson, An operational calculus for a class of abstract operator equations, J. Math. Anal. Appl. 37 (1972), 167–184. MR 291662, DOI 10.1016/0022-247X(72)90265-X
- Jerome A. Goldstein, Uniqueness for nonlinear Cauchy problems in Banach spaces, Proc. Amer. Math. Soc. 53 (1975), no. 1, 91–95. MR 377218, DOI 10.1090/S0002-9939-1975-0377218-5 J. A. Goldstein, Semigroups of operators and abstract Cauchy problems, Lecture Notes, Department of Mathematics, Tulane University, New Orleans, La., 1970.
- Einar Hille, Une généralisation du problème de Cauchy, Ann. Inst. Fourier (Grenoble) 4 (1952), 31–48 (1954) (French). MR 60731
- Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, Providence, R.I., 1957. rev. ed. MR 0089373
- J.-L. Lions, Opérateurs de transmutation singuliers et équations d’Euler Poisson Darboux généralisées, Rend. Sem. Mat. Fis. Milano 28 (1959), 124–137 (French). MR 114052, DOI 10.1007/BF02923018
- S. A. Tersenov, A singular Cauchy problem, Dokl. Akad. Nauk SSSR 196 (1971), 1032–1035 (Russian). MR 0276607
- Alexander Weinstein, On the wave equation and the equation of Euler-Poisson, Proceedings of Symposia in Applied Mathematics, Vol. V, Wave motion and vibration theory, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1954, pp. 137–147. MR 0063544
- Alexander Weinstein, The generalized radiation problem and the Euler-Poisson-Darboux equation, Summa Brasil. Math. 3 (1955), 125–147. MR 77770
- Alexandre Weinstein, Sur le problème de Cauchy pour l’équation de Poisson et l’équation des ondes, C. R. Acad. Sci. Paris 234 (1952), 2584–2585 (French). MR 49452
- Alexander Weinstein, Singular partial differential equations and their applications, Fluid Dynamics and Applied Mathematics (Proc. Sympos., Univ. of Maryland, 1961) Gordon and Breach, New York, 1962, pp. 29–49. MR 0153965
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 54 (1976), 149-153
- DOI: https://doi.org/10.1090/S0002-9939-1976-0390408-1
- MathSciNet review: 0390408