Díaz Díaz, Jesús Ildefonso and Casal, Alfonso C. and Vegas Montaner, José Manuel (2009) Blow-up in some ordinary and partial differential equations with time-delay. Dynamic Systems and Applications, 18 (1). pp. 29-46. ISSN 1056-2176
Restringido a Repository staff only hasta 31 December 2020.
Blow-up phenomena are analyzed for both the delay-differential equation (DDE) u'(t) = B'(t)u(t - tau), and the associated parabolic PDE (PDDE) partial derivative(t)u=Delta u+B'(t)u(t-tau,x), where B : [0, tau] -> R is a positive L(1) function which behaves like 1/vertical bar t - t*vertical bar(alpha), for some alpha is an element of (0, 1) and t* is an element of (0,tau). Here B' represents its distributional derivative. For initial functions satisfying u(t* - tau) > 0, blow up takes place as t NE arrow t* and the behavior of the solution near t* is given by u(t) similar or equal to B(t)u(t - tau), and a similar result holds for the PDDE. The extension to some nonlinear equations is also studied: we use the Alekseev's formula (case of nonlinear (DDE)) and comparison arguments (case of nonlinear (PDDE)). The existence of solutions in some generalized sense, beyond t = t* is also addressed. This results is connected with a similar question raised by A. Friedman and J. B. McLeod in 1985 for the case of semilinear parabolic equations.
|Uncontrolled Keywords:||ordinary and partial delay differential equations; parabolic partial differential equations; blow-up; Alekseev's formula.|
|Subjects:||Sciences > Mathematics > Differential equations|
S. Antontsev, J. I. Díaz, S. Shmarev, Energy methods for free boundary problems. Applications to nonlinear PDEs and Fluid Mechanics, Birkäuser, Boston, 2002.
Ph. Bénilan, M. G. Crandall and A. Pazy, Nonlinear evolution equations in Banach spaces. Book in preparation.
H. Brezis and T. Cazenave, A nonlinear heat equation with singular initial data. J. Anal. Math. 68 (1996), 277–304.
A. C. Casal, J. I. Díaz and J. M. Vegas, An extinction delay mechanism for abstract semilinear equations, XX CEDYA (XX Conference on Differential Equations and Applications, September 2007). Electronic Proceedings, Sevilla University.
C. Y. Chan, Blow-up and quenching phenomena due to concentrated nonlinear sources. Fifth International Conference on Dynamic Systems & Applications, May 2007. Proceedings of Dynamic Systems & Applications, Volume V (To appear). Dynamic Publishers. cf.
J. I. Díaz, Solutions with compact support for some degenerate parabolic problems. Nonlinear Analysis, Theory, Methods and Applications. Vol.3, No.6, 831 847, 1979.
J. I. Díaz, Nonlinear Partial Differential Equations and Free Boundaries. Vol.I. Elliptic equations. Research Notes in Mathematics no. 106, Pitman, London, 1985. A. Friedman, J. B. Mcleod, Blow-up of positive solutions of semilinear heat equation, Indiana Univ. Math. J. 34 (1985) 425–447.
A. F. Filippov, Differential equations with discontinuous righthand sides, Kluwer Academic, Dordrecht, 1988.
K. S. Ha: Nonlinear Functional evolutions in Banach spaces, Kluwer, AA Dordrecht, 2003.
J. K. Hale, Theory of functional differential equations, Springer, New York, 1977.
V. Laksmikantham, S. Leela, Differential an Integral Inequalities, vol I., Academic Press, New York, 1969.
R. H. Martin and H. L. Smith, Abstract Functional Differential Equations ad Reaction-Diffusion Systems, Trans. A.M.S., 231(1), (1990) 1–44.
C. V. Pao, Nonlinear parabolic and elliptic equations, Plenum, New York, 1992.
R. Redheffer and R. Redlinger, Quenching in time-delay systems: a summary and a counterexample, SIAM J. Math. Anal., 13, 1984, 1114–1124.
A. A. Samarski, V. A. Galaktionov,. S. P. Kurdyumov, and A. P. Mikhailov, Blow-up in quasilinear parabolic equations. Walter de Gruyter, Berlin, 1995.
H. L. Smith, Monotone semiflows generated by functional differential equations. Journal of Differential Equations 66, (1987), 420–442.
Ph. Souplet, Recent results and open problems on parabolic equations with gradient nonlinearities.Electronic Journal of Differential Equations, Vol. 2001(2001), No. 20, 1072–6691.
I. I. Vrabie, $C_0$C0-Semigroups and applications, North-Holland, Amsterdam, 2003.
W. Walter, Differential and Integral Inequalities, Springer-Verlag, Berlin-Heidelberg, 1970.
G. F. Webb, Functional differential equations and nonlinear semigroups in $L^p$Lp-spaces. J. Differential Equations 20 (1976), no. 1, 71–89.
J. Wu, Theory and applications of functional partial differential equations, Springer Verlag, New York, 1996.
|Deposited On:||08 May 2012 10:00|
|Last Modified:||06 Feb 2014 10:17|
Repository Staff Only: item control page