A first order differential inequality technique is used on suitably defined auxiliary functions to determine lower bounds for blow-up time in initial-boundary value problems for parabolic equations of the form ut = div ( ρ(u)gradu )+ f (u) if blow-up occurs. In addition, conditions which ensure that blow-up occurs or does not occur are presented. © 2007 Elsevier Inc. All rights reserved.

This paper deals with a degenerate parabolic equation vt = ∆v + av1 ∥v∥1 α1 subject to homogeneous Dirichlet condition. The local existence of a nonnegative weak solution is given. The blow-up and global existence conditions of nonnegative solutions are obtained. Moreover, we establish the precise blow-up rate estimates for all the blow-up solutions.

Five types of blow-up patterns that can occur for the 4th-order semilinear parabolic equation of reaction-diffusion type ut = −∆2u+ |u|p−1u in R × (0, T ), p > 1, limt→T− supx∈RN |u(x, t)| = +∞, are discussed. For the semilinear heat equation ut = ∆u+ u , various blow-up patterns were under scrutiny since 1980s, while the case of higher-order diffusion was studied much less, regardless a wide r...

In the paper, several problems on the periodic Degasperis-Procesi equation with weak dissipation are investigated. At first, the local well-posedness of the equation is established by Kato’s theorem and a precise blow-up scenario of the solutions is given. Then, several criteria guaranteeing the blow-up of the solutions are presented. Moreover, the blow-up rate and blow-up set of the blowing-up...

We consider the 1D nonlinear Schrödinger equation (NLS) with focusing point nonlinearity, (0.1) i∂tψ + ∂ 2 xψ + δ|ψ|p−1ψ = 0, where δ = δ(x) is the delta function supported at the origin. In the L supercritical setting p > 3, we construct self-similar blow-up solutions belonging to the energy space Lx ∩Ḣ x. This is reduced to finding outgoing solutions of a certain stationary profile equation. ...

For some deterministic nonlinear PDEs on the torus whose solutions may blow up in finite time, we show that, under suitable conditions term, blow-up is delayed by multiplicative noise of transport type a certain scaling limit. The main result applied to 3D Keller–Segel, Fisher–KPP, and 2D Kuramoto–Sivashinsky equations, yielding long-time existence for large initial data with high probability.