scholarly journals ON THE BLOW-UP OF THE NON-LOCAL THERMISTOR PROBLEM

2007 ◽  
Vol 50 (2) ◽  
pp. 389-409 ◽  
Author(s):  
N. I. Kavallaris ◽  
T. Nadzieja
Keyword(s):  
Blow Up ◽  
Critical Value ◽  
Lower Solution ◽  
Initial Condition ◽  
Robin Problem ◽  
Thermistor Problem ◽  
Decay Condition ◽  
Blowing Up ◽  
Non Local ◽  
The Neumann Problem

AbstractThe conditions under which the solution of the non-local thermistor problem\begin{gather*} u_t=\Delta u+\frac{\lambda f(u)}{(\int_{\varOmega}f(u)\,\mathrm{d}x)^{2}},\quad x\in\varOmega\subset\mathbb{R}^N,\ N\geq2,\ t>0, \\ \frac{\partial u(x,t)}{\partial\nu}+\beta(x)u(x,t)=0,\quad x\in\partial\varOmega,\ t>0, \\ u(x,0)=u_0(x),\quad x\in\varOmega, \end{gather*}blows up are investigated. We assume that $f(s)$ is a decreasing function and that it is integrable in $(0,\infty)$. Considering a suitable functional we prove that for all $\lambda\gt0$ the solution of the Neumann problem blows up in finite time. The same result is obtained for the Robin problem under the assumption that $\lambda$ is sufficiently large $(\lambda\gg 1)$. In the proof of existence of blow-up for the Dirichlet problem we use the subsolution technique. We are able to construct a blowing-up lower solution under the assumption that either $\lambda\gt\lambda^*$ or $0\lt\lambda\lt\lambda^*$, for some critical value $\lambda^*$, and that the initial condition is sufficiently large provided also that $f(s)$ satisfies the decay condition $\int_0^\infty[sf(s)-s^2f'(s)]\,\mathrm{d} s\lt\infty$.

scholarly journals ESTIMATES OF BLOW-UP TIME FOR A NON-LOCAL REACTIVE–CONVECTIVE PROBLEM MODELLING OHMIC HEATING OF FOODS

2006 ◽  
Vol 49 (1) ◽  
pp. 215-239 ◽  
Author(s):  
C. V. Nikolopoulos ◽  
D. E. Tzanetis
Keyword(s):  
Steady State ◽  
Initial Data ◽  
Blow Up ◽  
Ohmic Heating ◽  
Steady State Solution ◽  
Critical Value ◽  
Stationary Solutions ◽  
State Solution ◽  
Non Local ◽  
The Difference

AbstractIn this work, we estimate the blow-up time for the non-local hyperbolic equation of ohmic type, $u_t+u_{x}=\lambda f(u)/(\int_{0}^1f(u)\,\mathrm{d} x)^{2}$, together with initial and boundary conditions. It is known that, for $f(s)$, $-f'(s)$ positive and $\int_0^\infty f(s)\,\mathrm{d} s\lt\infty$, there exists a critical value of the parameter $\lambda>0$, say $\lambda^\ast$, such that for $\lambda>\lambda^\ast$ there is no stationary solution and the solution $u(x,t)$ blows up globally in finite time $t^\ast$, while for $\lambda\leq\lambda^\ast$ there exist stationary solutions. Moreover, the solution $u(x,t)$ also blows up for large enough initial data and $\lambda\leq\lambda^\ast$. Thus, estimates for $t^\ast$ were found either for $\lambda$ greater than the critical value $\lambda^\ast$ and fixed initial data $u_0(x)\geq0$, or for $u_0(x)$ greater than the greatest steady-state solution (denoted by $w_2\geq w^*$) and fixed $\lambda\leq\lambda^\ast$. The estimates are obtained by comparison, by asymptotic and by numerical methods. Finally, amongst the other results, for given $\lambda$, $\lambda^*$ and $0\lt\lambda-\lambda^*\ll1$, estimates of the following form were found: upper bound $\epsilon+c_1\ln[c_2(\lambda-\lambda^*)^{-1}]$; lower bound $c_3(\lambda-\lambda^*)^{-1/2}$; asymptotic estimate $t^\ast\sim c_4(\lambda-\lambda^\ast)^{-1/2}$ for $f(s)=\mathrm{e}^{-s}$. Moreover, for $0\lt\lambda\leq\lambda^*$ and given initial data $u_0(x)$ greater than the greatest steady-state solution $w_2(x)$, we have upper estimates: either $c_5\ln(c_6A^{-1}_0+1)$ or $\epsilon+c_7\ln(c_8\zeta^{-1})$, where $A_0$, $\zeta$ measure, in some sense, the difference $u_0-w_2$ (if $u_0\to w_2+$, then $A_0,\zeta\to0+$). $c_i\gt0$ are some constants and $0\lt\epsilon\ll1$, $0\ltA_0,\zeta$. Some numerical results are also given.

Estimates of blow-up time for a non-local problem modelling an Ohmic heating process

2002 ◽  
Vol 13 (3) ◽  
pp. 337-351 ◽  
Author(s):  
N. I. KAVALLARIS ◽  
C. V. NIKOLOPOULOS ◽  
D. E. TZANETIS
Keyword(s):  
Blow Up ◽  
Ohmic Heating ◽  
Numerical Estimate ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Stationary Solutions ◽  
Upper And Lower Bounds ◽  
Heating Process ◽  
Non Local ◽  
Initial Boundary Value

We consider an initial boundary value problem for the non-local equation, ut = uxx+λf(u)/(∫1-1f (u)dx)2, with Robin boundary conditions. It is known that there exists a critical value of the parameter λ, say λ*, such that for λ > λ* there is no stationary solution and the solution u(x, t) blows up globally in finite time t*, while for λ < λ* there exist stationary solutions. We find, for decreasing f and for λ > λ*, upper and lower bounds for t*, by using comparison methods. For f(u) = e−u, we give an asymptotic estimate: t* ∼ tu(λ−λ*)−1/2 for 0 < (λ−λ*) [Lt ] 1, where tu is a constant. A numerical estimate is obtained using a Crank-Nicolson scheme.

EXPLOSIVE INSTABILITY DUE TO 3-WAVE OR 4-WAVE MIXING, WITH OR WITHOUT DISSIPATION

2008 ◽  
Vol 06 (04) ◽  
pp. 413-428 ◽  
Author(s):  
HARVEY SEGUR
Keyword(s):  
Initial Data ◽  
Nonlinear Waves ◽  
Finite Time ◽  
Blow Up ◽  
Wave Mixing ◽  
Explosive Instability ◽  
Effective Threshold ◽  
Blowing Up ◽  
Gain Energy

It is known that an "explosive instability" can occur when nonlinear waves propagate in certain media that admit 3-wave mixing. In that context, three resonantly interacting wavetrains all gain energy from a background source, and all blow up together, in finite time. A recent paper [17] showed that explosive instabilities can occur even in media that admit no 3-wave mixing. Instead, the instability is caused by 4-wave mixing, and results in four resonantly interacting wavetrains all blowing up in finite time. In both cases, the instability occurs in systems with no dissipation. This paper reviews the earlier work, and shows that adding a common form of dissipation to the system, with either 3-wave or 4-wave mixing, provides an effective threshold for blow-up. Only initial data that exceed the respective thresholds blow up in finite time.

scholarly journals Solutions with Prescribed Local Blow-up Surface for the Nonlinear Wave Equation

2019 ◽  
Vol 19 (4) ◽  
pp. 639-675
Author(s):  
Thierry Cazenave ◽  
Yvan Martel ◽  
Lifeng Zhao
Keyword(s):  
Wave Equation ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Blow Up ◽  
Finite Energy ◽  
Energy Solution ◽  
Compact Set ◽  
Existence Of A Solution ◽  
Blowing Up ◽  
Finite Energy Solution

AbstractWe prove that any sufficiently differentiable space-like hypersurface of {{\mathbb{R}}^{1+N}} coincides locally around any of its points with the blow-up surface of a finite-energy solution of the focusing nonlinear wave equation {\partial_{tt}u-\Delta u=|u|^{p-1}u} on {{\mathbb{R}}\times{\mathbb{R}}^{N}}, for any {1\leq N\leq 4} and {1<p\leq\frac{N+2}{N-2}}. We follow the strategy developed in our previous work (2018) on the construction of solutions of the nonlinear wave equation blowing up at any prescribed compact set. Here to prove blow-up on a local space-like hypersurface, we first apply a change of variable to reduce the problem to blowup on a small ball at {t=0} for a transformed equation. The construction of an appropriate approximate solution is then combined with an energy method for the existence of a solution of the transformed problem that blows up at {t=0}. To obtain a finite-energy solution of the original problem from trace arguments, we need to work with {H^{2}\times H^{1}} solutions for the transformed problem.

scholarly journals On Blowing Up the Pokies: The Pokie Lounge as a Cultural Site of Neoliberal Governmentality in Australia

2011 ◽  
Vol 17 (2) ◽  
Author(s):  
Fiona Nicoll
Keyword(s):  
Blow Up ◽  
Public Affairs ◽  
Cultural Dynamics ◽  
Neoliberal Governmentality ◽  
Television Shows ◽  
Self Help ◽  
Blowing Up ◽  
Fatal Attraction ◽  
Commercial Radio ◽  
Gaming Machines

In 1999 The Whitlams, a popular ‘indie’ band named after a former Australian prime minister whose government was controversially sacked in 1975 by the Governor-General, released a single titled ‘Blow up the Pokies’. Written about a former band member’s fatal attraction to electronic gaming machines (henceforth referred to as ‘pokies’), the song was mixed by a top LA producer, a decision that its writer and The Whitlam’s front-man, Tim Freedman, describes as calculated to ‘get it on big, bombastic commercial radio’. The investment paid off and the song not only became a big hit for the band, it developed a legacy beyond the popular music scene, with Freedman invited to write the foreword of a ‘self-help manual for giving up gambling’ as well as appearing on public affairs television shows to discuss the issue of problem gambling. The lyrics of ‘Blow up the Pokies’ frame the central themes of this article: spaces, technologies and governmentality of gambling. It then explores what cultural articulations of resistance to the pokie lounge tell us about broader social and cultural dynamics of neoliberal governmentality in Australia.

scholarly journals Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up

2019 ◽  
Vol 2019 (754) ◽  
pp. 225-251 ◽  
Author(s):  
James Isenberg ◽  
Haotian Wu
Keyword(s):  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Blow Up ◽  
Curvature Flow ◽  
Type Ii ◽  
Matched Asymptotics ◽  
Curve Shortening Flow ◽  
Initial Hypersurface ◽  
Rate Dependent ◽  
Blowing Up

Abstract We study the phenomenon of Type-II curvature blow-up in mean curvature flows of rotationally symmetric noncompact embedded hypersurfaces. Using analytic techniques based on formal matched asymptotics and the construction of upper and lower barrier solutions enveloping formal solutions with prescribed behavior, we show that for each initial hypersurface considered, a mean curvature flow solution exhibits the following behavior near the “vanishing” time T: (1) The highest curvature concentrates at the tip of the hypersurface (an umbilic point), and for each choice of the parameter {\gamma>\frac{1}{2}} , there is a solution with the highest curvature blowing up at the rate {(T-t)^{{-(\gamma+\frac{1}{2})}}} . (2) In a neighborhood of the tip, the solution converges to a translating soliton which is a higher-dimensional analogue of the “Grim Reaper” solution for the curve-shortening flow. (3) Away from the tip, the flow surface approaches a collapsing cylinder at a characteristic rate dependent on the parameter γ.

Sign in / Sign up

Export Citation Format

Share Document