Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-22T18:50:52.524Z Has data issue: false hasContentIssue false
  • English
  • Français

Quasi-linear parabolic partial differential equations with delays in the highest order spatial derivatives

Part of: Parabolic equations and systems

Published online by Cambridge University Press:  09 April 2009

Jiongmin Yong  and
Liping Pan
Jiongmin Yong
Affiliation:
Department of MathematicsFudan UniversityShanghai 200433, China
Liping Pan
Affiliation:
Department of MathematicsFudan UniversityShanghai 200433, China
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A class of functional differential equations in some Hilbert space are studied. The results are applicable to many quasi-linear parabolic paratial differential equations with (possibly) countably many discrete delays and finitely many distributed delays in the highest order spatial derivatives. For the linear case, an evolution operator on the underline space H is introduced, via which a variation of constant formula for the solution of the equation in the underline space H is derived. Some spectral properties of the generator of the solution semigroup defined on some appropriate space are discussed as well.

Keywords

Functional differential equationdelay differential equation.

MSC classification

Secondary: 35K99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 54 , Issue 2 , April 1993 , pp. 174 - 203
Copyright
Copyright © Australian Mathematical Society 1993

References

[1]Ardito, A. and Ricciardi, P., ‘Existence and regularity for linear delay partial differential equations’, Nonlinear Anal. 4 (1980), 411–414.CrossRefGoogle Scholar
[2]Ardito, A. and Vernole, P., ‘A delay parabolic Cauchy-Dirichiet problem in interpolation space’, Nonlinear Anal. 9 (1985), 445–454.CrossRefGoogle Scholar
[3]Di Blasio, G., Kunisch, K. and Sinestrari, E., ‘L2-regularity for parabolic partial integrodifferential equations with delay in the highest-order derivatives’, J. Math. Anal. Appl. 102 (1984), 38–57.CrossRefGoogle Scholar
[4]Di Blasio, G., Kunisch, K. and Sinestrari, E., ‘Stability for abstract linear functional differential equations’, Israel J. Math. 50 (1985), 231–263.CrossRefGoogle Scholar
[5]Datko, R., ‘Representation of solutions and stability of linear differential-difference equalions in a Banach space’, J. Differential Eqn. 29 (1978), 105–166.CrossRefGoogle Scholar
[6]Delfour, M. C. and Mitter, S. D., ‘Hereditary differential systems with constant delays, I. general case’, J. Differential Eqn. 12 (1972), 213–235.CrossRefGoogle Scholar
[7]Delfour, M. C. and Mitter, S. D., ‘Hereditary differential systems with constant delays, II. a class of affine systems and the adjoint problem’, J. Differential Eqn. 18 (1975) 18–28.CrossRefGoogle Scholar
[8]Hale, J., Theory of functional differential equations, (Springer, New York, 1977).CrossRefGoogle Scholar
[9]Kunisch, K., ‘A semigroup approach to partial differential equations with delays’, in Abstract Cauchy problems and functional differential equations (eds. Kappel, F. and Schappacher, W.), Research Notes in Mathematics 48 (Pitman, Boston, 1981) pp. 53–70.Google Scholar
[10]Kunisch, K. and Schappacher, W., ‘Variation of constants formulas for partial differential equations with delays’, Nonlinear Anal. 5 (1989), 123–142.CrossRefGoogle Scholar
[11]Kunisch, K. and Schappacher, W., ‘Necessary conditions for partial differential equations with delay to generate C0-semigroups’, J. Differential Eqn. 50 (1983), 49–79.CrossRefGoogle Scholar
[12]Kunisch, K. and Schappacher, W., ‘Mild and strong solutions for partial differential equations with delay’, Ann. Math. Pura. Appl. 125 (1980), 193–219.CrossRefGoogle Scholar
[13]Lions, J. L. and Magenes, E., Nonhomogeneous boundary value problems and applications II (Springer, New York, 1972).Google Scholar
[14]Nakagiri, S., ‘On the fundamental solution of delay-differential equations in Banach spaces’, J. Differential Eqn. 41 (1981), 349–368.CrossRefGoogle Scholar
[15]Nakagiri, S., ‘Structural properties of functional differential equations in Banach spaces,’ Osaka J. Math. 25 (1988), 353–398.Google Scholar
[16]Pazy, A., Semigroups of linear operators and applications to partial differential equations (Springer, New York, 1983).CrossRefGoogle Scholar
[17]Da Prato, G. and Grisvard, P., ‘Equations d'évolution abstraites non linéaires de type paraboliqué’, Ann. Mat. Pura. Appl. 120 (1979), 329–396.CrossRefGoogle Scholar
[18]Sinestrari, E. and Vernole, P., ‘Retarded functional parabolic equations in interpolation spaces’, Nonlinear Anal. 3 (1979), 583–593.CrossRefGoogle Scholar
[19]Travis, C. and Webb, G., ‘Partial differential equations with deviating arguments in the time variables’, J. Math. Anal. Appl. 56 (1976), 397–409.CrossRefGoogle Scholar
[20]Triebel, H., Interpolation theory, function spaces, differential operators (North-Holland, Amsterdam, 1978).Google Scholar
[21]Yosida, K., Functional Analysis (Springer, New York, 1980).Google Scholar
[22]You, Y., ‘On generator of solution semigroup for linear retarded evolution equation in M2 space’, J. Fudan Univ. Natur. Sci. 21 (1982), 164–173.Google Scholar