We derive conditions on the initial data, including cases where the initial momentum density is not of one sign, that produce blow-up of the induced solution to the modified integrable Camassa-Holm equation with cubic nonlinearity. The blow-up conditions are formulated in terms of the initial momentum density and the average initial energy.

We study the anisotropic flow V = bk where b > 0. We prove that if 13 < σ ≤ 1, only type I blow up occurs and if 0 < σ < 13 Type II blow up occurs. We also establish a prior estimates and existence of stationary solutions of self-similar curves and the existence of “ Abresch-Langer” type curves.

In this paper, usng the gluing formula of Gromov-Witten invariants under symplectic cutting, due to Li and Ruan, we studied the Gromov-Witten invariants of blow-ups at a smooth point or along a smooth curve. We established some relations between Gromov-Witten invariants of M and its blow-ups at a smooth point or along a smooth curve.

In this paper we establish new nonlinear Liouville theorems for parabolic problems on half spaces. Based on the Liouville theorems, we derive estimates for the blow-up of positive solutions of indefinite parabolic problems and investigate the complete blow-up of these solutions. We also discuss a priori estimates for indefinite elliptic problems.

We study the blow-up behavior for positive solutions of a reaction–diffusion equationwith nonnegative variable coefficient. When there is no stationary solution, we show that the solution blows up in finite time. Under certain conditions, we then show that any point with zero source cannot be a blow-up point. © 2012 Elsevier Ltd. All rights reserved.

In this article, we study the higher-order semilinear parabolic equation ut + (−∆)u = |u|, (t, x) ∈ R+ × R , u(0, x) = u0(x), x ∈ R . Using the test function method, we derive the blow-up critical exponent. And then based on integral inequalities, we estimate the life span of blow-up solutions.

In this paper we study the blow up problem for positive solutions of parabolic and hyperbolic problems with reaction terms of local and nonlocal type involving a variable exponent. We prove the existence of initial data such that the corresponding solutions blow up at a finite time.

We study solutions of some supercritical parabolic equations which blow up in finite time but continue to exist globally in the weak sense. We show that the minimal continuation becomes regular immediately after the blow-up time and if it blows up again, it can only do so finitely many times.

In this paper, we draw upon Islamic teachings to address two questions. How do ethics deepen and advance our understanding of the whistleblowing act? To what extent are promoted in practice? First, have undertaken a thematic content analysis holy book Qur’an, supported by Sunnah (Prophetic Traditions). This has yielded novel ethics-based framework comprising five aspects process: What should on...