Critical Quenching Exponents for Heat Equations Coupled with Nonlinear Boundary Flux

We discuss the quenching phenomena for a system of heat equations coupled with nonlinear boundary flux. We determine a critical value for the exponents in the boundary flux, such that only in the super critical case the simultaneous quenching can happen for any solution.     

Non-simultaneous quenching for a slow diffusion system coupled at the boundary

This paper deals with the quenching behavior of positive solutions to the Newton filtration equations coupled with boundary singularities.We determine quenching rates for nonsimultaneous quenching at f...     

Numerical quenching of a system of equations coupled at the boundary

We study numerical approximations of solutions of the following system of heat equations, coupled at the boundary through a nonlinear flux: where p and q are parameters. We prove that the solutions of a semidiscretization in space quench in finite time. Moreover, we describe in terms of p and q the simultaneous versus non‐simultaneous quenching phenomena. We also find the numerical extinction sets. Finally, in order to obtain the correct quenching rates in the non‐simultaneous case we present some adaptive methods. Copyright © 2009 John Wiley & Sons, Ltd.     

Incomplete quenching of heat equations with coupled singular logarithms boundary fluxes

This paper deals with quenching phenomena for a heat equations with coupled singular logarithms boundary fluxes. Under appropriate hypotheses, the non-simultaneous quenching of the solution for the system is proved, and the estimates of quenching rates are given. Then we give a natural continuation of the solution (u,v) after the quenching time when the equations occurs non-simultaneous quenching. Moreover, we identify the heat equations verified by the continuation beyond quenching time, i.e., the equations occurs incomplete quenching.     

Quenching Versus Blow-up

This paper is concerned with the semilinear heat equation u_t = Δu - u^{-q} in Ω × (0, T) under the nonlinear boundary condition \frac{∂u}{∂v} = u^p on ∂Ω × (0, T). Criteria for finite time quenching and blow-up are established, quenching and blow-up sets are discussed, and the rates of quenching and blow-up are obtained.     

Incomplete quenching in a system of heat equations coupled at the boundary

In this paper we find a possible continuation for quenching solutions to a system of heat equations coupled at the boundary condition. This system exhibits simultaneous and non-simultaneous quenching. For non-simultaneous quenching our continuation is a solution of a parabolic problem with Neumann boundary conditions. We also give some results for simultaneous quenching and present some numerical experiments that suggest that the approximations are not uniformly bounded in this case.     

Local and non‐local boundary quenching

The quenching problem is examined for a one‐dimensional heat equation with a non‐linear boundary condition that is of either local or non‐local type. Sufficient conditions are derived that establish both quenching and non‐quenching behaviour. The growth rate of the solution near quenching is also given for a power‐law non‐linearity. The analysis is conducted in the context of a nonlinear Volterra integral equation that is equivalent to the initial–boundary value problem. Copyright © 1999 John Wiley & Sons, Ltd.     

Non-simultaneous quenching in a system of heat equations coupled at the boundary

We study the solutions of a parabolic system of heat equations coupled at the boundary through a nonlinear flux. We characterize in terms of the parameters involved when non-simultaneous quenching may appear. Moreover, if quenching is non-simultaneous we find the quenching rate, which surprisingly depends on the flux associated to the other component.     

A complete classification for non-simultaneous blow-up

This work deals with heat equations coupled via nonlinear boundary flux which obey different laws. We give a complete classification for non-simultaneous and simultaneous blow-up by covering all of the possible exponents.     

QUENCHING ON BOUNDARY TO THE NEWTON FILTRATION EQUATION(Ⅰ)

This paper discusses the global existence and quenching of the solution tothe Newton filtration equation with the nonlinear boundary condition. The authors alsodiscuss the profile of the quenching solution in the quenching time and obtain the quenchingrate of the quenching solution.     

A complete classification of simultaneous blow-up rates

We study the simultaneous blow-up rates of a system of two heat equations coupled through the boundary in a nonlinear way. We complete the previous known results by covering the whole range of possible parameters.     

Non-simultaneous quenching in a system of heat equations coupled at the boundary

We study the solutions of a parabolic system of heat equations coupled at the boundary through a nonlinear flux. We characterize in terms of the parameters involved when non-simultaneous quenching may appear. Moreover, if quenching is non-simultaneous we find the quenching rate, which surprisingly depends on the flux associated to the other component. Partially supported by project BFM2002-04572 (Spain). Partially supported by UBA grant EX046, CONICET and Fundación Antorchas (Argentina). Received: February 17, 2004; revised: July 5, 2004     

Asymptotic properties of singular solutions in degenerate parabolic equation with boundary flux

ABSTRACT

This paper deals with blow-up and quenching solutions of degenerate parabolic problem involving m-Laplacian operator and nonlinear boundary flux. The blow-up and quenching criteria are classified under the conditions on the initial data but with less conditions on the relationship among the exponents, respectively. Moreover, asymptotic properties including singular rates, set and time estimates are determined for the blow-up solutions and the quenching solutions, respectively.     

QUENCHING ON BOUNDARY TO THE NEWTON FILTRATION EQUATION (I)

This paper discusses the global existence and quenching of the solution to the Newton filtration equation with the nonlinear boundary condition. The authors also discuss the profile of the quenching solution in the quenching time and obtain the quenching rate of the quenching solution.     

Non-simultaneous versus simultaneous quenching in a coupled nonlinear parabolic system

This paper deals with finite-time quenching for the nonlinear parabolic system with coupled singular absorptions: u_t=Δu−v^−p, v_t=Δv−u^−q in Ω×(0,T) subject to positive Dirichlet boundary conditions, where p,q>0, Ω is a bounded domain in with smooth boundary. We obtain the sufficient conditions for global existence and finite-time quenching of solutions, and then determine the blow-up of time-derivatives and the quenching set for the quenching solutions. As the main results of the paper, a very clear picture is obtained for radial solutions with Ω=B_R: the quenching is simultaneous if p,q≥1, and non-simultaneous if p<1≤q or q<1≤p; if p,q<1 with , then both simultaneous and non-simultaneous quenching may happen, depending on the initial data. In determining the non-simultaneous quenching criteria of the paper, some new ideas have been introduced to deal with the coupled singular inner absorptions and inhomogeneous Dirichlet boundary value conditions, in addition to techniques frequently used in the literature.     

Total versus single point blow-up in a localized heat system

This paper considers a heat system with localized sources and local couplings subject to null Dirichlet boundary conditions, for which both total and single point blow-up are possible. The aim of the paper is to identify the total and single point blow-up via a complete classification for all the nonlinear parameters in the model. As preliminaries of the paper, simultaneous versus non-simultaneous blow-up of solutions is involved, too. The results are then compared with those for another kind of heat system coupled via localized sources in a previous paper of the authors.     

高压气体淬火过程中的温度场与相变耦合问题的计算

高压气体淬火技术是一种现代的、有效的材料加工技术．在Cheng所得到的高压气体淬火过程中非线性表面换热系数的基础上,用有限单元法对钢淬火过程的温度场与相变耦合问题进行了模拟计算．在数值计算中,材料的热物性系数被处理为温度和相变体积百分比的函数．为避免数值解的震荡,采用了Norsette有理近似法．     

Blow-up estimates for system of heat equations coupled via nonlinear boundary flux

This paper deals with a system of heat equations coupled via nonlinear boundary flux. The precise blow-up rate estimates are established together with the blow-up set. It is observed that there is some quantitative relationship regarding the blow-up properties between the heat system with coupled nonlinear boundary flux terms and the corresponding reaction–diffusion system with the same nonlinear terms as the source.     

New explicit approximate solution of MHD viscoelastic boundary layer flow over stretching sheet

In this paper, the study the momentum and heat transfer characteristics in an incompressible electrically conducting non‐Newtonian boundary layer flow of a viscoelastic fluid over a stretching sheet. The partial differential equations governing the flow and heat transfer characteristics are converted into highly nonlinear coupled ordinary differential equations by similarity transformations. The resultant coupled highly nonlinear ordinary differential equations are solved by means of, homotopy analysis method (HAM) for constructing an approximate solution of heat transfer in magnetohydrodynamic (MHD) viscoelastic boundary layer flow over a stretching sheet with non‐uniform heat source. The proposed method is a strong and easy to use analytic tool for nonlinear problems and does not need small parameters in the equations. The HAM solutions contain an auxiry parameter, which provides a convenient way of controlling the convergence region of series solutions. The results obtained here reveal that the proposed method is very effective and simple for solving nonlinear evolution equations. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in physics. Copyright © 2012 John Wiley & Sons, Ltd.     

Quenching Time Estimates for Semilinear Parabolic Equations Controlled by Two Absorption Sources in Control System

This paper deals with the quenching solution of the initial boundary value problem for aclass of semilinear reaction-diffusion equation controlled by two absorption sources in control system and estimate upper bound and lower bound of the quenching time. We point that the number of absorption sources influences the time of quenching phenomenon.The solution can solve some boundary value problem in control system.