# Existence of the fundamental solution of a second order evolution equation

Annales Polonici Mathematici (1997)

- Volume: 66, Issue: 1, page 15-35
- ISSN: 0066-2216

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topJan Bochenek. "Existence of the fundamental solution of a second order evolution equation." Annales Polonici Mathematici 66.1 (1997): 15-35. <http://eudml.org/doc/269952>.

@article{JanBochenek1997,

abstract = {We give sufficient conditions for the existence of the fundamental solution of a second order evolution equation. The proof is based on stable approximations of an operator A(t) by a sequence $\{A_n(t)\}$ of bounded operators.},

author = {Jan Bochenek},

journal = {Annales Polonici Mathematici},

keywords = {evolution problem; stable family of operators; stable approximations of the evolution operator; fundamental solution; Cauchy problem; uniformly correct Cauchy problem; fundamental solutions; second-order evolution equations; Cauchy problems},

language = {eng},

number = {1},

pages = {15-35},

title = {Existence of the fundamental solution of a second order evolution equation},

url = {http://eudml.org/doc/269952},

volume = {66},

year = {1997},

}

TY - JOUR

AU - Jan Bochenek

TI - Existence of the fundamental solution of a second order evolution equation

JO - Annales Polonici Mathematici

PY - 1997

VL - 66

IS - 1

SP - 15

EP - 35

AB - We give sufficient conditions for the existence of the fundamental solution of a second order evolution equation. The proof is based on stable approximations of an operator A(t) by a sequence ${A_n(t)}$ of bounded operators.

LA - eng

KW - evolution problem; stable family of operators; stable approximations of the evolution operator; fundamental solution; Cauchy problem; uniformly correct Cauchy problem; fundamental solutions; second-order evolution equations; Cauchy problems

UR - http://eudml.org/doc/269952

ER -

## References

top- [1] J. Bochenek and T. Winiarska, Evolution equations with parameter in the hyperbolic case, Ann. Polon. Math. 64 (1996), 47-60. Zbl0855.34070
- [2] H. O. Fattorini, Ordinary differential equations in linear topological spaces, I, J. Differential Equations 5 (1968), 72-105. Zbl0175.15101
- [3] H. O. Fattorini, Ordinary differential equations in linear topological spaces, II, J. Differential Equations 6 (1969), 50-70. Zbl0181.42801
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- [5] T. Kato, Perturbation Theory for Linear Operators, Grundlehren Math. Wiss. 132, Springer, New York, 1980.
- [6] M. Kozak, A fundamental solution of a second-order differential equation in a Banach space, Univ. Iagel. Acta Math. 32 (1995), 275-289. Zbl0855.34073
- [7] S. Krein, Linear Differential Equations in Banach Space, Amer. Math. Soc., 1972.
- [8] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci. 44, Springer, 1983.
- [9] H. Tanabe, Equations of Evolution, Pitman, London, 1979.
- [10] C. C. Travis and G. F. Webb, Cosine families and abstract nonlinear second order differential equations, Acta Math. Acad. Sci. Hungar. 32 (1978), 75-96. Zbl0388.34039
- [11] T. Winiarska, Evolution equations of second order with operator depending on t, in: Selected Problems of Mathematics, Cracow University of Technology, Anniversary issue, 1995, 299-311.
- [12] K. Yosida, Functional Analysis, Springer, New York, 1980.

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