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 rates for heat equations with coupled singular nonlinear boundary flux

We study finite time quenching for heat equations coupled via singular nonlinear bound-ary flux. A criterion is proposed to identify the simultaneous and non-simultaneous quenchings. In particular, three kinds of simultaneous quenching rates are obtained for different nonlinear exponent re-gions and appropriate initial data. This extends an original work by Pablo, Quir′os and Rossi for a heat system with coupled inner absorption terms subject to homogeneous Neumann boundary conditions.     

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.     

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.     

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 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.     

Quenching for a reaction–diffusion system with logarithmic singularity

In this paper, we study the quenching phenomenon for a reaction–diffusion system with singular logarithmic source terms and positive Dirichlet boundary conditions. Some sufficient conditions for quenching of the solutions in finite time are obtained, and the blow-up of time-derivatives at the quenching point is verified. Furthermore, under appropriate hypotheses, the non-simultaneous quenching of the system is proved, and the estimates of quenching rate is given.     

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.     

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.     

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．     

THE SINGLE-POINT QUENCHING OF NONLINEAR DEGENERATE FUNCTIONAL REACTION-DIFFUSION EQUATION

This article is concerned with the quenching phenomena of the nonlinear degenerate functional reaction-diffusion equation. Some results are obtained on the single-point quenching and the uniqueness of quenching.     

QUENCHING PROBLEMS OF DEGENERATE FUNCTIONAL REACTION-DIFFUSION EQUATION

This paper is concerned with the quenching problem of a degenerate functional reaction-diffusion equation. The quenching problem and global existence of solution for the reaction-diffusion equation are derived and, some results of the positive steady state solutions for functional elliptic boundary value are also presented.     

Optimization of technology parameters for the plane-strain component in the process of gas quenching

The paper introduces an optimization method for the technology parameters of the plane-strain component in the process of gas quenching. Distortion, residual stress, average surface hardness and standard deviation of surface hardness are regarded as the optimization objectives. A new heat transfer coefficient model is presented, five distinct heat transfer coefficients are used at various regions of the model. The five heat transfer coefficients are regarded as the design variables, and four regressive equations are established by using response surface method. The four equations, respectively represent the relations between the four optimization objectives and the design variables. A multi-objectives optimization model is established, and the multi-objectives optimization model is optimized by the non-linear method. The optimized technology parameters are used to simulate the gas quenching process by FEM software. The quenching results after optimization are compared with those before optimization. The comparison shows that the quenching quality after optimization is better than that before optimization. After optimization, the four optimization objectives are all improved.     

反应-扩散方程的熄灭问题

In this article, we consider a reaction-diffusion differential equation with initial value conditions u(x, 0) = 0 on [0, a] and boundary condition ux + αiu = 0 on Γ= {0, a} × (0, T), and the quenching happens for the reaction-diffusion equation.     

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.     

Quenching phenomenon of a time‐fractional diffusion equation with singular source term

In this paper, a time‐fractional diffusion equation with singular source term is considered. The Caputo fractional derivative with order 0<α ?1 is applied to the temporal variable. Under specific initial and boundary conditions, we find that the time‐fractional diffusion equation presents quenching solution that is not globally well‐defined as time goes to infinity. The quenching time is estimated by using the eigenfunction of linear fractional diffusion equation. Moreover, by implementing a finite difference scheme, we give some numerical simulations to demonstrate the theoretical analysis. Copyright © 2017 John Wiley & Sons, Ltd.     

ON THE CAUCHY PROBLEM FOR A REACTION-DIFFUSION SYSTEM WITH SINGULAR NONLINEARITY

We consider the growth rate and quenching rate of the following problem with singular nonlinearityfor some positive constants b：, b2 （see Theorem 3.3 for the parametersfor some constantsHence, the solution （u, v） quenches at the originx = 0 at the same time ＇1＇ （see Theorem 4.3）. We also tind various other conditions tor the solution to quench in a finite time and obtain the corresponding decay rate of the solution near the quenching time.     

一类带反应项的多种群趋化性模型解的性态(英文)

本文研究了一类带反应的多种群趋化性Othmer-Stevens模型的问题,利用抛物方程上下解和迭代的方法,获得了该模型在不同条件下的解的爆破,熄灭等渐进性态,推广了单种群模型的结果。     

退化的半线性反应扩散方程组解的熄灭

In this paper,the initial boundary value problem of semilinear degenerate reaction-diffusion systems is studied.The regularization method and upper and lower solutions technique are employed to show the existence and continuation of a positive classical solution.The location of quenching points is found.The critical length is estimated by the eigenvalue method.     

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.