scholarly journals Finite time blow-up and global solutions for a nonlocal parabolic equation with Hartree type nonlinearity

2020 ◽  
Vol 19 (6) ◽  
pp. 3093-3112
Author(s):  
Xiaoliang Li ◽  
◽  
Baiyu Liu ◽  
Keyword(s):  
Parabolic Equation ◽  
Finite Time ◽  
Blow Up ◽  
Global Solutions ◽  
Nonlocal Parabolic Equation

On the blow-up of solutions of a convective reaction diffusion equation

1993 ◽  
Vol 123 (3) ◽  
pp. 433-460 ◽  
Author(s):  
J. Aguirre ◽  
M. Escobedo
Keyword(s):  
Positive Solutions ◽  
Finite Time ◽  
Blow Up ◽  
Global Solutions ◽  
Reaction Diffusion ◽  
Scalar Function ◽  
Reaction Diffusion Equation ◽  
Semilinear Parabolic Equation ◽  
The Cauchy Problem ◽  
Image Position

SynopsisWe study the blow-up of positive solutions of the Cauchy problem for the semilinear parabolic equationwhere u is a scalar function of the spatial variable x ∈ ℝN and time t > 0, a ∈ ℝV, a ≠ 0, 1 < p and 1 ≦ q. We show that: (a) if p > 1 and 1 ≦ q ≦ p, there always exist solutions which blow up in finite time; (b) if 1 < q ≦ p ≦ min {1 + 2/N, 1 + 2q/(N + 1)} or if q = 1 and 1 < p ≦ l + 2/N, then all positive solutions blow up in finite time; (c) if q > 1 and p > min {1 + 2/N, 1 + 2q/N + 1)}, then global solutions exist; (d) if q = 1 and p > 1 + 2/N, then global solutions exist.

Global solutions and finite time blow-up for fourth order nonlinear damped wave equation

2018 ◽  
Vol 59 (6) ◽  
pp. 061503 ◽  
Author(s):  
Runzhang Xu ◽  
Xingchang Wang ◽  
Yanbing Yang ◽  
Shaohua Chen
Keyword(s):  
Wave Equation ◽  
Fourth Order ◽  
Finite Time ◽  
Blow Up ◽  
Global Solutions ◽  
Damped Wave Equation ◽  
Nonlinear Damped Wave Equation

Finite time blow-up of complex solutions of the conserved Kuramoto–Sivashinsky equation in ℝd and in the torus 𝕋d, d ⩾ 1

2019 ◽  
Vol 149 (5) ◽  
pp. 1175-1188
Author(s):  
Léo Agélas
Keyword(s):  
Sobolev Space ◽  
Initial Data ◽  
Besov Space ◽  
Finite Time ◽  
Blow Up ◽  
Global Solutions ◽  
Large Initial Data ◽  
Complex Solutions ◽  
Complex Valued ◽  
Sivashinsky Equation

AbstractWe consider complex-valued solutions of the conserved Kuramoto–Sivashinsky equation which describes the coarsening of an unstable solid surface that conserves mass and that is parity symmetric. This equation arises in different aspects of surface growth. Up to now, the problem of existence and smoothness of global solutions of such equations remained open in ℝd and in the torus 𝕋d, d ⩾ 1. In this paper, we answer partially to this question. We prove the finite time blow-up of complex-valued solutions associated with a class of large initial data. More precisely, we show that there is complex-valued initial data that exists in every Besov space (and hence in every Lebesgue and Sobolev space), such that after a finite time, the complex-valued solution is in no Besov space (and hence in no Lebesgue or Sobolev space).

scholarly journals Large Solutions for Semilinear Parabolic Equations Involving Some Special Classes of Nonlinearities

2010 ◽  
Vol 2010 ◽  
pp. 1-11 ◽  
Author(s):  
Constantin P. Niculescu ◽  
Ionel Rovenţa
Keyword(s):  
Parabolic Equation ◽  
Parabolic Equations ◽  
Blow Up ◽  
Neumann Boundary ◽  
Semilinear Parabolic Equations ◽  
New Class ◽  
Large Solutions ◽  
Nonlocal Parabolic Equation ◽  
Work Done ◽  
Semilinear Parabolic

We consider a new class of nonlinearities for which a nonlocal parabolic equation with Neumann boundary conditions has finite time blow-up solutions. Our approach is inspired by previous work done by Jazar and Kiwan (2008) and El Soufi et al. (2007).

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