scholarly journals Single point blow-up and final profile for a perturbed nonlinear heat equation with a gradient and a non-local term

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Bouthaina Abdelhedi ◽  
Hatem Zaag
Keyword(s):  
Heat Equation ◽  
Blow Up ◽  
Single Point ◽  
Nonlinear Heat Equation ◽  
Non Local ◽  
Final Profile ◽  
Local Term

scholarly journals Global blow-up for a semilinear heat equation on a subspace

2015 ◽  
Vol 145 (5) ◽  
pp. 893-923 ◽  
Author(s):  
C. J. Budd ◽  
J. W. Dold ◽  
V. A. Galaktionov
Keyword(s):  
Heat Equation ◽  
Initial Function ◽  
Blow Up ◽  
Neumann Boundary ◽  
Compact Set ◽  
Sharp Estimates ◽  
Semilinear Heat Equation ◽  
Non Local ◽  
Image Position ◽  
Local Term

We study the asymptotic behaviour as t → T–, near a finite blow-up time T > 0, of decreasing-in-x solutions to the following semilinear heat equation with a non-local term:with Neumann boundary conditions and strictly decreasing initial function u0(x) with zero mass. We prove sharp estimates for u(x, t) as t → T–, revealing a non-uniform global blow-up:uniformly on any compact set [δ, 1], δ ∈ (0, 1).

Global blow-up of a nonlinear heat equation

1986 ◽  
Vol 104 (1-2) ◽  
pp. 161-167 ◽  
Author(s):  
A. A. Lacey
Keyword(s):  
Heat Equation ◽  
Parabolic Equations ◽  
Finite Time ◽  
Positive Measure ◽  
Blow Up ◽  
Nonlinear Heat Equation ◽  
Semilinear Parabolic Equations ◽  
Semilinear Parabolic

SynopsisSolutions to semilinear parabolic equations of the form ut = Δu + f(u), x in Ω, which blow up at some finite time t* are investigated for “slowly growing” functions f. For nonlinearities such as f(s) = (2 +s)(ln(2 +s))1+b with 0 < b < l,u becomes infinite throughout Ω as t→t* −. It is alsofound that for marginally more quickly growing functions, e.g. f(s) = (2 + s)(ln(2 +s))2, u is unbounded on some subset of Ω which has positive measure, and is unbounded throughout Ω if Ω is a small enough region.

Local well‐posedness and blow‐up for an inhomogeneous nonlinear heat equation

2020 ◽  
Vol 43 (8) ◽  
pp. 5264-5272
Author(s):  
Rasha Alessa ◽  
Aisha Alshehri ◽  
Haya Altamimi ◽  
Mohamed Majdoub
Keyword(s):  
Heat Equation ◽  
Blow Up ◽  
Nonlinear Heat Equation ◽  
Well Posedness

scholarly journals ASYMPTOTIC BEHAVIOUR AND BLOW-UP FOR A NONLINEAR DIFFUSION PROBLEM WITH A NON-LOCAL SOURCE TERM

2004 ◽  
Vol 47 (2) ◽  
pp. 375-395 ◽  
Author(s):  
N. I. Kavallaris
Keyword(s):  
Steady State ◽  
Blow Up ◽  
Spatial Dimension ◽  
Diffusion Problem ◽  
Local Source ◽  
Primary 35K60 ◽  
Non Local ◽  
Steady State Problem ◽  
Steady State Solutions ◽  
Local Term

AbstractIn this work, the behaviour of solutions for the Dirichlet problem of the non-local equation$$ u_t=\varDelta(\kappa(u))+\frac{\lambda f(u)}{(\int_{\varOmega}f(u)\,\mathrm{d}x)^p},\quad \varOmega\subset\mathbb{R}^N,\quad N=1,2, $$is studied, mainly for the case where $f(s)=\mathrm{e}^{\kappa(s)}$. More precisely, the interplay of exponent $p$ of the non-local term and spatial dimension $N$ is investigated with regard to the existence and non-existence of solutions of the associated steady-state problem as well as the global existence and finite-time blow-up of the time-dependent solutions $u(x,t)$. The asymptotic stability of the steady-state solutions is also studied.AMS 2000 Mathematics subject classification: Primary 35K60. Secondary 35B40

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