Quasilinear evolution equations with
operators dependent on $t$

T. Winiarska

The purpose of this paper is to present some theorems on existence and uniqueness of solutions for some semilinear Cauchy problems of second order with operators $A(t)$ not densely defined in a given Banach space $X$. To this end, we begin with reduction of our problem to a~problem in which the operators have the same (independent of $t$) domain $D$.

1. Bochenek J. Existence of the fundamental solution of a second order evolution equation, Ann. Polon. Math. 66 (1997) 15–35.

2. Fattorini H. O. Ordinary differential equations in linear topological spaces, Intern. J. Diff. Equations, 5 (1968), 72–105.

3. Fattorini H. O. Ordinary differential equations in linear topological spaces, Intern. J. Diff. Equations, 6 (1969), 50–70.

4. Kozak M. A fundamental solution of a second order differential equation in a Banach space, Univ. Iagel. Acta Math. 32 (1995), 275–289.

5. Kozak M. An abstract linear second order temporally inhomogeneous differential equation II, Univ. Iagel. Acta Math. 32 (1995), 263–274.

6. Krein S. Linear Differential Equations in Banach Space, Amer. Math. Soc., 1972.

7. von Neerven J. The adjoint of a semigroup of Linear Operators, Springer–Verlag, 1992.

8. Pazy A. Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1983.

9. Radoń M. Application of the extrapolation spaces in theory of differential equations in Banach space, PhD thesis, 2002 (in Polish).

10. Travis C. C., Webb G. F. Cosine family and abstract nonlinear second order differential equation, Acta Math. Hungar. 32 (1978), 75–96.

11. Winiarska T. Extrapolation Banach spaces and abstract semilinear second order differential equations, Inter. J. Differ. Equat. and Appl. 6 (2002), No. 4, 449–465.

12. Winiarska T., Winiarski T. Class of regularity of domains paramaterized families of closed linear operators, Ann. Polon. Math. 80 (2003), 231–241.

	Quasilinear evolution equations with operators dependent on $t$
Author	T. Winiarska twiniars@usk.pk.edu.pl Institute of Mathematics, Cracow University of Technology,Warszawska 24, 31–155 Cracow, Poland
Abstract	The purpose of this paper is to present some theorems on existence and uniqueness of solutions for some semilinear Cauchy problems of second order with operators $A(t)$ not densely defined in a given Banach space $X$. To this end, we begin with reduction of our problem to a~problem in which the operators have the same (independent of $t$) domain $D$.
Keywords	quasilinear evolution equation, Banach space, Cauchy problem
DOI	doi:10.30970/ms.21.2.170-178
Reference	1. Bochenek J. Existence of the fundamental solution of a second order evolution equation, Ann. Polon. Math. 66 (1997) 15–35. 2. Fattorini H. O. Ordinary differential equations in linear topological spaces, Intern. J. Diff. Equations, 5 (1968), 72–105. 3. Fattorini H. O. Ordinary differential equations in linear topological spaces, Intern. J. Diff. Equations, 6 (1969), 50–70. 4. Kozak M. A fundamental solution of a second order differential equation in a Banach space, Univ. Iagel. Acta Math. 32 (1995), 275–289. 5. Kozak M. An abstract linear second order temporally inhomogeneous differential equation II, Univ. Iagel. Acta Math. 32 (1995), 263–274. 6. Krein S. Linear Differential Equations in Banach Space, Amer. Math. Soc., 1972. 7. von Neerven J. The adjoint of a semigroup of Linear Operators, Springer–Verlag, 1992. 8. Pazy A. Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1983. 9. Radoń M. Application of the extrapolation spaces in theory of differential equations in Banach space, PhD thesis, 2002 (in Polish). 10. Travis C. C., Webb G. F. Cosine family and abstract nonlinear second order differential equation, Acta Math. Hungar. 32 (1978), 75–96. 11. Winiarska T. Extrapolation Banach spaces and abstract semilinear second order differential equations, Inter. J. Differ. Equat. and Appl. 6 (2002), No. 4, 449–465. 12. Winiarska T., Winiarski T. Class of regularity of domains paramaterized families of closed linear operators, Ann. Polon. Math. 80 (2003), 231–241.
Pages	170-178
Volume	21
Issue	2
Year	2004
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Quasilinear evolution equations with operators dependent on $t$