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Blow-up in some ordinary and partial differential equations with time-delay

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

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Abstract

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.

Item Type:Article
Uncontrolled Keywords:ordinary and partial delay differential equations; parabolic partial differential equations; blow-up; Alekseev's formula.
Subjects:Sciences > Mathematics > Differential equations
ID Code:15136
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