Journal of the Mathematical Society of Japan

Degenerate integrodifferential equations of parabolic type with Robin boundary conditions: $\bm{L^2}$ -theory

Abstract

This paper is devoted to solving a degenerate parabolic integrodifferential equation with the Robin boundary condition. We begin with solving the equation without the integral delay term. For that purpose we introduce some new unknown function following Favini and Yagi [4] and construct the fundamental solution to the equation to be satisfied by it by the method of Kato and Tanabe [5]. Using this fundamental solution we transform the original problem to an easily solvable integral equation for the time derivative of the new unknown function.

Article information

Source
J. Math. Soc. Japan Volume 61, Number 1 (2009), 133-176.

Dates
First available: 9 February 2009

http://projecteuclid.org/euclid.jmsj/1234189031

Digital Object Identifier
doi:10.2969/jmsj/06110133

Mathematical Reviews number (MathSciNet)
MR2272874

Zentralblatt MATH identifier
05530294

Citation

FAVINI, Angelo; LORENZI, Alfredo; TANABE, Hiroki. Degenerate integrodifferential equations of parabolic type with Robin boundary conditions: L 2 -theory. Journal of the Mathematical Society of Japan 61 (2009), no. 1, 133--176. doi:10.2969/jmsj/06110133. http://projecteuclid.org/euclid.jmsj/1234189031.

References

• M. V. Bulatov, Integro-differential systems with a degenerate matrix multiplying the derivative, Diff. Eqs., 38 (2002), 731–737, (Russian, Diff. Uravn., 38 (2002), 692–697).
• M. G. Crandall and J. A. Nohel, An abstract functional differential equation and a related nonlinear Volterra equation, Israel J. Math., 29 (1978), 313–328.
• A. Favini, A. Lorenzi and H. Tanabe, Degenerate integrodifferential equations of parabolic type, Differential Equations Inverse and Direct Problems, A series of Lecture Notes in Pure and Applied Mathematics, 251, (eds. A. Favini and A. Lorenzi), Chapman & Hall/CRC, Taylor & Francis Group, Boca Raton-London-New York, 2006, pp.,91–109.
• A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, Marcel Dekker, Inc., New York-Basel-Hong Kong, 1999.
• T. Kato and H. Tanabe, On the abstract evolution equation, Osaka Math. J., 14 (1962), 107–133.
• A. Lorenzi and H. Tanabe, Inverse and direct problems for nonautonomous degenerate integrodifferential equations of parabolic type with Dirichlet boundary conditions, Differential Equations Inverse and Direct Problems, A series of Lecture Notes in Pure and Applied Mathematics, 251, (eds. A. Favini and A. Lorenzi), Chapman & Hall/CRC, Taylor & Francis Group, Boca Raton-London-New York, 2006, pp.,197–243.
• J. Prüss, On resolvent operators for linear integrodifferential equations of Volterra type, J. Integral Equations, 5 (1983), 211–236.