Dynamic Systems and Applications xx (200x) xx-xx BLOW-UP IN SOME ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS WITH TIME-DELAY

Blow-up phenomena are analyzed for both the delay-differential equation (DDE) u(t) = B(t)u(t− τ), and the associated parabolic PDE (PDDE) ∂tu = ∆u+B (t)u(t− τ, x), where B : [0, τ ] → R is a positive L function which behaves like 1/ |t− t∗| , for some α ∈ (0, 1) and t∗ ∈ (0, τ). Here B′ represents its distributional derivative. For initial functions satisfying u(t∗ − τ) > 0, blow up takes place as t↗ t∗ and the behavior of the solution near t∗ is given by u(t) ' B(t)u(t− τ), 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. AMS (MOS) Subject Classification. 34K05, 34K12, 34K40, 35B05, 35B30, 35B40, 35B60, 35B65, 35K55.