Blow-up in some ordinary and partial differential equations with time-delay, Dynamic Systems and Applications

ABSTRACT. 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 L1 function which behaves like 1 / |t − t∗| α, 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 ր 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