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.

First we give a truly short proof of the major blow up result [Si] on higher dimensional semilinear wave equations. Using this new method, we also establish blow up phenomenon for wave equations with a potential. This complements the recent interesting existence result by [GHK], where the blow up problem was left open.

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.

In the present work, we establish an optimal estimate for the electric potential difference between closely adjacent spherical perfect conductors in n dimensional space (n ≥ 2). This result indicates that electric fields blow up as a pair of spherical perfect conductors approach each other, and provides the lower bound with the optimal blowup rate which was recently established by Bao, Li and Y...

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...

Abstract Let D be an open set in $$\mathbb {R}^d$$ ℝ d and $$u:D\to \mathbb {R}$$ u : D → a (non-negative) local minimizer of $$\mathcal F_\Lambda $$ ℱ...

The present paper is concerned with the Cauchy problem { ∂tu = ∆u + u in R × (0,∞), u(x, 0) = u0(x) ≥ 0 in R , with p,m > 1. A solution u with bounded initial data is said to blow up at a finite time T if lim supt↗T ‖u(t)‖L∞(RN ) = ∞. For N ≥ 3 we obtain, in a certain range of values of p, weak solutions which blow up at several times and become bounded in intervals between these blow-up times....

In this paper we consider the slightly L-supercritical gKdV equations ∂tu + (uxx + u|u|)x = 0, with the nonlinearity 5 < p < 5 + ε and 0 < ε ≪ 1 . In the previous paper [10] we know that there exists an stable selfsimilar blow-up dynamics for slightly L-supercritical gKdV equations. Such solution can be viewed as solutions with single blow-up point. In this paper we will prove the existence of ...

A CAD paradigm is presented to produce blow moulds for Poly Ethylene Terephthalate (PET) bottles decorated with ornamental patterns. This work integrates design and manufacturing of decorated blow moulds which, in turn, are utilized to blow PET bottles. The parametric geometric modeling techniques are employed to customize a CAD system, tailoring to suit the requirements of the bottle making in...