Complete second order differential equations in Banach spaces with dynamic boundary conditions

Journal of Differential Equations

Volumn 200, Issue 1, 2004, Pages 105-136

  Xiao, Ti Jun ^a,b   Liang, Jin ^a,b  

^a UNIVERSITY OF SCIENCE AND TECHNOLOGY OF CHINA   (China)

^b UNIVERSITY OF TÜBINGEN   (Germany)

Author keywords

Differential equations in Banach spaces; Dynamic boundary conditions; Hyperbolic equations; Inhomogeneous; Mixed initial boundary value problem; Second order; Wellposedness; Wentzell boundary conditions

Indexed keywords

EID: 2342590155     PISSN: 00220396     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.jde.2004.01.011     Document Type: Article

References (27)

1. K.T. Andrews, K.L. Kuttler, M. Shillor, Second order evolution equations with dynamic boundary conditions, J. Math. Anal. Appl. 197 (1996) 781-795.
2. R.W. Carroll, R.E. Showalter, Singular and Degenerate Cauchy Problems, Academic Press, New York, 1976.
3. V. Casarino, K.J. Engel, R. Nagel, G. Nickel, A semigroup approach to boundary feedback systems, Integral Equations Operator Theory 47 (2003) 289-306.
4. J. Escher, Quasilinear parabolic systems with dynamical boundary conditions, Comm. Partial Differential Equations 18 (1993) 1309-1364.
5. H.O. Fattorini, The Cauchy Problem, Encyclopedia of Mathematics and its Applications, Vol. 18, Addison-Wesley, London, 1983.
6. H.O. Fattorini, Second Order Linear Differential Equations in Banach Spaces, Elsevier Science Publishers B.V., Amsterdam, 1985.
7. o-semigroups generated by second order differential operators with general Wentzell boundary conditions, Proc. Amer. Math. Soc. 128 (2000) 1981-1989.
8. A. Favini, G.R. Goldstein, J.A. Goldstein, S. Romanelli, The one dimensional wave equation with Wentzell boundary condition, in: S. Aicovici, N. Pavel (Eds.), Differential Equations and Control Theory, Lecture Notes in Pure Applied Mathematics, Vol. 225, Marcel Dekker, New York, 2001, pp. 139-145. Tübinger Berichte zur Funktionalanalysis 9 (1999/2000) 162-168.
9. A. Favini, G. R. Goldstein, J.A. Goldstein, S. Romanelli, The heat equation with generalized Wentzell boundary condition, J. Evolution Equations 2 (2002) 1-19.
10. A. Favini, E. Obrecht, Conditions for parabolicity of second order abstract differential equations, Differential Integral Equations 4 (1991) 1005-1020.
11. C.L. Frota, J.A. Goldstein, Some nonlinear wave equations with acoustic boundary conditions, J. Differential Equations 164 (2000) 92-109.
12. C.G. Gal, G.R. Goldstein, J.A. Goldstein, Oscillatory boundary conditions for acoustic wave equations, J. Evolution Equations, to appear.
13. G. Greiner, Perturbing the boundary conditions of a generator, Houston J. Math. 13 (1987) 213-229.
14. M. Grobbelaar-Van Dalsen, N. Sauer, Dynamic boundary conditions for the Navier-Stokes equations, Proc. Roy. Soc. Edinburgh Sect. A 113 (1989) 1-11.
15. T. Hintermann, Evolution equations with dynamic boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A 113 (1989) 43-60.
16. M. Kramar, D. Mugnolo, R. Nagel, Theory and applications of one-sided coupled operator matrices, Confer. Sem. Mat. Univ. Bari No. 283 (2002) 1-29.
17. J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation, J. Differential Equations 50 (1983) 163-182.
18. J.L. Lions, Equations differentielles operationelles et problemes aux limites, Springer, Berlin, 1961.
19. J.L. Lions, E. Magenes, Non-homogeneous Boundary Value Problems, and Applications, (I), Springer, New York, 1970.
20. R.E. Showalter, Degenerate evolution equations and applications, Indiana Univ. Math. J. 23 (1974) 655-677.
21. H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978.
22. T.J. Xiao, On the well-posedness of the Cauchy problem for a kind of complete second order differential equations, J. Sichuan Univ. (Sichuan Daxue Xuebao) 25 (1988) 421-428.
23. T.J. Xiao, J. Liang, On complete second order linear differential equations in Banach spaces, Pacific J. Math. 142 (1990) 175-195.
24. T.J. Xiao, J. Liang, Differential operators and C-wellposedness of complete second order abstract Cauchy problems, Pacific J. Math. 186 (1998) 167-200.
25. T.J. Xiao, J. Liang, The Cauchy Problem for Higher Order Abstract Differential Equations, Lecture Notes in Mathematics 1701, Springer, Berlin, New York, 1998.
26. T.J. Xiao, J. Liang, Second order parabolic equations in Banach spaces with dynamic boundary conditions, preprint, 2002.
27. T.J. Xiao, J. Liang, A solution to an open problem for wave equations with generalized Wentzell boundary conditions, Math. Ann. 327 (2003) 351-363.