On an instantaneous blow-up of solutions of evolutionary problems on the half-line

2018 ◽  
Vol 82 (5) ◽  
pp. 914-930 ◽  
Author(s):  
M. O. Korpusov
Keyword(s):  
Blow Up ◽  
Half Line

scholarly journals An extension of the method of brackets. Part 1

2017 ◽  
Vol 15 (1) ◽  
pp. 1181-1211 ◽  
Author(s):  
Ivan Gonzalez ◽  
Karen Kohl ◽  
Lin Jiu ◽  
Victor H. Moll
Keyword(s):  
Efficient Method ◽  
Blow Up ◽  
Divergent Series ◽  
Half Line ◽  
Definite Integrals ◽  
Small Collection ◽  
Formal Representations

Abstract The method of brackets is an efficient method for the evaluation of alarge class of definite integrals on the half-line. It is based on a small collection of rules, some of which are heuristic. The extension discussed here is based on the concepts of null and divergent series. These are formal representations of functions, whose coefficients an have meromorphic representations for n ∈ ℂ, but might vanish or blow up when n ∈ ℕ. These ideas are illustrated with the evaluation of a variety of entries from the classical table of integrals by Gradshteyn and Ryzhik.

scholarly journals A nonlinear diffusion equation with reaction localized in the half-line

2021 ◽  
Vol 4 (3) ◽  
pp. 1-24
Author(s):  
Raúl Ferreira ◽  
◽  
Arturo de Pablo ◽  
Keyword(s):  
Global Existence ◽  
Diffusion Equation ◽  
Heat Equation ◽  
Nonlinear Diffusion ◽  
Blow Up ◽  
Nonlinear Diffusion Equation ◽  
Half Line ◽  
Blow Up Rate ◽  
Global Reaction

<abstract><p>We study the behaviour of the solutions to the quasilinear heat equation with a reaction restricted to a half-line</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ u_t = (u^m)_{xx}+a(x) u^p, $\end{document} </tex-math></disp-formula></p> <p>$ m, p &gt; 0 $ and $ a(x) = 1 $ for $ x &gt; 0 $, $ a(x) = 0 $ for $ x &lt; 0 $. We first characterize the global existence exponent $ p_0 = 1 $ and the Fujita exponent $ p_c = m+2 $. Then we pass to study the grow-up rate in the case $ p\le1 $ and the blow-up rate for $ p &gt; 1 $. In particular we show that the grow-up rate is different as for global reaction if $ p &gt; m $ or $ p = 1\neq m $.</p></abstract>

Singularity formation in the one-dimensional supercooled Stefan problem

1996 ◽  
Vol 7 (2) ◽  
pp. 119-150 ◽  
Author(s):  
Miguel A. Herrero ◽  
Juan J. L. Velázquez
Keyword(s):  
Stefan Problem ◽  
Finite Time ◽  
Blow Up ◽  
Half Line ◽  
Singularity Formation ◽  
Asymptotics Of Solutions ◽  
One Dimensional ◽  
The One

It is well-known that solutions to the one-dimensional supercooled Stefan problem (SSP) may exhibit blow-up in finite time. If we consider (SSP) in a half-line with zero flux conditions at t = 0, blow-up occurs if there exists T < ∞ such that limt↑Ts(t) > 0 and lim inft↑T⋅(t) = – ∞,s(t) being the interface of the problem under consideration. In this paper, we derive the asymptotics of solutions and interfaces near blow-up. We shall also use these results to discuss the possible continuation of solutions beyond blow-up.

Positive solutions for a quasilinear system Via blow up

1993 ◽  
Vol 18 (12) ◽  
pp. 2071-2106
Author(s):  
Philippe Clément ◽  
Raúl Manásevich ◽  
Enzo Mitidieri
Keyword(s):  
Positive Solutions ◽  
Blow Up ◽  
Quasilinear System
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