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.     

Blow-up phenomena for a system of semilinear heat equations with nonlinear boundary flux

This paper deals with the blow-up for a system of semilinear r-Laplace heat equations with nonlinear boundary flux. It is shown that, under certain conditions on the nonlinearities and data, blow-up will occur at some finite time, and when blow-up does occur upper and lower bounds for the blow-up time are obtained.     

Critical Fujita exponents for degenerate parabolic equations coupled via nonlinear boundary flux

We establish the critical Fujita exponents for degenerate parabolic equations coupled via nonlinear boundary flux and then determine the blow-up rates and the blow-up sets for the nonglobal solutions.     

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.     

Multiple blow-up rates in a coupled heat system with mixed-type nonlinearities

This paper studies heat equations with inner absorptions and coupled boundary fluxes of mixed-type nonlinearities. At first, the critical exponent is obtained, and simply described via a characteristic algebraic system introduced by us. Then, as the main results of the paper, three blow-up rates are established under different dominations of nonlinearities for the one-dimensional case, and represented in another characteristic algebraic system. In particular, it is observed that unlike those in previous literature on parabolic models with absorptions, two of the multiple blow-up rates obtained here do depend on the absorption exponents. In the known works, the absorptions affect the blow-up criteria, the blow-up time, as well as the initial data required for the blow-up of solutions, all without changing the blow-up rates. To our knowledge, this is the first example of absorption-dependent blow-up rates, exploiting the significant interactions among diffusions, inner absorptions and nonlinear boundary fluxes in the coupled system. It is also proved that the blow-up of solutions in the model occurs on the boundary only.     

耦合热方程组解的多重 Blow-up 速率

This paper deals with a heat system coupled via local and localized sources subject to null Dirichlet boundary conditions. Based on a complete classification for all the four nonlinear parameters, we establish multiple blow-up rates for the system under various dominations. We also determine uniform blow-up profiles for the three cases where localized source couplings dominate the system.     

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.     

A Reaction-diffusion system with nonlinear absorption terms and boundary flux

This paper deals with a reaction-diffusion system with nonlinear absorption terms and boundary flux. As results of interactions among the six nonlinear terms in the system, some sufficient conditions on global existence and finite time blow-up of the solutions are described via all the six nonlinear exponents appearing in the six nonlinear terms. In addition, we also show the influence of the coefficients of the absorption terms as well as the geometry of the domain to the global existence and finite time blow-up of the solutions for some cases. At last, some numerical results are given.     

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.     

A semilinear parabolic system with generalized Wentzell boundary condition

In this paper, we consider a weak coupled semilinear parabolic system with general Wentzell boundary condition. We prove the well-posedness of the problem and derive different conditions in terms of the powers of the nonlinear terms under which the global solution exists and finite time blow-up occurs.     

Blow-up vs. spurious steady solutions

In this paper, we study the blow-up problem for positive solutions of a semidiscretization in space of the heat equation in one space dimension with a nonlinear flux boundary condition and a nonlinear absorption term in the equation. We obtain that, for a certain range of parameters, the continuous problem has blow-up solutions but the semidiscretization does not and the reason for this is that a spurious attractive steady solution appears.

     

Simultaneous and non-simultaneous blow-up for a cross-coupled parabolic system

This paper deals with a parabolic system, cross-coupled via a nonlinear source and a nonlinear boundary flux. We get a necessary and sufficient condition for the existence of non-simultaneous blow-up. In particular, four different simultaneous blow-up rates are obtained in different regions of parameters, described by an introduced characteristic algebraic system. It is observed that different initial data may result in different simultaneous blow-up rates even in the same region of parameters.     

具有吸收和非线性边界流的非线性扩散方程的爆破估计

本文研究一类具有非线性吸收和非线性边界流的非线性扩散方程，建立了解的爆破速率估计，所得结果依赖于模型中三种非线性机制之间的相互作用。     

Asymptotic analysis of a coupled nonlinear parabolic system

This paper deals with asymptotic analysis of a parabolic system with inner absorptions and coupled nonlinear boundary fluxes. Three simultaneous blow-up rates are established under different dominations of nonlinearities, and simply represented in a characteristic algebraic system introduced for the problem. In particular, it is observed that two of the multiple blow-up rates are absorption-related. This is substantially different from those for nonlinear parabolic problems with absorptions in all the previous literature, where the blow-up rates were known as absorptionindependent. The results of the paper rely on the scaling method with a complete classification for the nonlinear parameters of the model. The first example of absorption-related blow-up rates was recently proposed by the authors for a coupled parabolic system with mixed type nonlinearities. The present paper shows that the newly observed phenomena of absorptionrelated blow-up rates should be due to the coupling mechanism, rather than the mixed type nonlinearities.      

一类交叉耦合非线性反应扩散方程组的爆破分析

主要研究了具有混合型的多重非线性项的抛物方程组的初边值问题.方程组中的非线性项是幂函数和指数混合型的.这些非线性项组合出了源-流交叉耦合,通过比较原理得到了方程组的上下解,并得到了解有限时刻爆破的临界指标.     

Global solutions and blow-up problems for a nonlinear degenerate parabolic system coupled via nonlocal sources

This paper concerns with a nonlinear degenerate parabolic system coupled via nonlocal sources, subjecting to homogeneous Dirichlet boundary condition. The main aim of this paper is to study conditions on the global existence and/or blow-up in finite time of solutions, and give the estimates of blow-up rates of blow-up solutions.     

Blow-up sets for linear diffusion equations in one dimension

We consider the heat equation in the half-line with Dirichlet boundary data which blow up in finite time. Though the blow-up set may be any interval [0,a], a ? [0,￥]a\in[0,\infty] depending on the Dirichlet data, we prove that the effective blow-up set, that is, the set of points x 3 0x\ge0 where the solution behaves like u(0,t), consists always only of the origin. As an application of our results we consider a system of two heat equations with a nontrivial nonlinear flux coupling at the boundary. We show that by prescribing the non-linearities the two components may have different blow-up sets. However, the effective blow-up sets do not depend on the coupling and coincide with the origin for both components.