Necessary and sufficient conditions for complete blow-up and extinction for one-dimensional quasilinear heat equations

1995 ◽  
Vol 129 (3) ◽  
pp. 225-244 ◽  
Author(s):  
Victor A. Galaktionov ◽  
Juan L. Vazquez
Keyword(s):  
Blow Up ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Heat Equations ◽  
One Dimensional ◽  
Necessary And Sufficient

GENERALIZED DECIMATION INVARIANT SEQUENCES IN DIMENSION N

2002 ◽  
Vol 12 (04) ◽  
pp. 709-737 ◽  
Author(s):  
A. BARBÉ ◽  
F. VON HAESELER
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
One Dimensional ◽  
Necessary And Sufficient

We generalize the concept of one-dimensional decimation invariant sequences, i.e. sequences which are invariant under a specific rescaling, to dimension N. After discussing the elementary properties of decimation-invariant sequences, we focus our interest on their periodicity. Necessary and sufficient conditions for the existence of periodic decimation invariant sequences are presented.

scholarly journals A classification of explosions in dimension one

2009 ◽  
Vol 29 (2) ◽  
pp. 715-731 ◽  
Author(s):  
E. SANDER ◽  
J. A. YORKE
Keyword(s):  
Sufficient Conditions ◽  
Global Bifurcation ◽  
Necessary And Sufficient Conditions ◽  
Homoclinic Tangency ◽  
Discontinuous Change ◽  
One Dimensional ◽  
One Dimensional Maps ◽  
Dimension One ◽  
Necessary And Sufficient

AbstractA discontinuous change in the size of an attractor is the most easily observed type of global bifurcation. More generally, anexplosionis a discontinuous change in the set of recurrent points. An explosion often results from heteroclinic and homoclinic tangency bifurcations. We prove that, for one-dimensional maps, explosions are generically the result of either tangency or saddle-node bifurcations. Furthermore, we give necessary and sufficient conditions for generic tangency bifurcations to lead to explosions.

On multi-dimensional annealing problems

1989 ◽  
Vol 105 (1) ◽  
pp. 177-184 ◽  
Author(s):  
Terence Chan
Keyword(s):  
Continuous Function ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
One Dimensional ◽  
Dimensional Diffusion ◽  
Deterministic Function ◽  
Image Position ◽  
Necessary And Sufficient

In [1] Chan and Williams considered a one-dimensional diffusion of the formwhere F is a strictly increasing continuous function with F(0) = 0 and ε is a decreasing deterministic function such that ε(0) is finite and ε(t) ↓ 0 as t↑ ∞, and gave necessary and sufficient conditions for Yt →0 a.s. as t→∞.

scholarly journals A Nonlocal Operator Breaking the Keller–Osserman Condition

2017 ◽  
Vol 17 (4) ◽  
pp. 715-725 ◽  
Author(s):  
Raúl Ferreira ◽  
Mayte Pérez-Llanos
Keyword(s):  
Blow Up ◽  
Existence Of Solutions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Nonlocal Operator ◽  
Large Solutions ◽  
Necessary And Sufficient ◽  
Semilinear Problem

AbstractThis work is concerned about the existence of solutions to the nonlocal semilinear problem\left\{\begin{aligned} &\displaystyle{-}\int_{{\mathbb{R}}^{N}}J(x-y)(u(y)-u(x% ))\,dy+h(u(x))=f(x),&&\displaystyle x\in\Omega,\\ &\displaystyle u=g,&&\displaystyle x\in{\mathbb{R}}^{N}\setminus\Omega,\end{% aligned}\right.verifying that {\lim_{x\to\partial\Omega,\,x\in\Omega}u(x)=+\infty}, known in the literature as large solutions. We find out that the relation between the diffusion and the absorption term is not enough to ensure such existence, not even assuming that the boundary datum g blows up close to {\partial\Omega}. On the contrary, the role to obtain large solutions is played only by the interior source f, which gives rise to large solutions even without the presence of the absorption. We determine necessary and sufficient conditions on f providing large solutions and compute the blow-up rates of such solutions in terms of h and f. Finally, we also study the uniqueness of large solutions.

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