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     

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...     

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.     

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.     

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.     

Non-simultaneous blow-up in coupled heat equations with multi-nonlinearities

This paper studies coupled heat equations with multi-nonlinearities of six nonlinear parameters. The critical blow-up exponent is established via a complete classification for all the six nonlinear parameters, where a precise analysis on the geometry of Ω and the absorption coefficients is given for the balanced interaction situation among the multi-nonlinearities. The main attention is contributed to non-simultaneous phenomena in the model to determine the necessary and sufficient conditions of non-simultaneous blow-up with suitable initial data, as well as the conditions under which any blow-up must be non-simultaneous. Finally, the model of the paper is characterized by a comparison with the known results for other models.     

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.     

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.     

Non-simultaneous blowup in heat equations with nonstandard growth conditions

This paper cares about blowup solutions for a system of n-componential heat equations coupled via localized reactions and with variable exponents. The criteria for non-simultaneous and simultaneous blowup are established for radial solutions with or without assumptions on initial data, including the existence of non-simultaneous blowup for n components; any blowup must be simultaneous or non-simultaneous.     

Blow‐up analysis for a system of heat equations coupled via nonlinear boundary conditions

In this paper, we study a system of heat equations coupled via nonlinear boundary conditions (1) Here p, q>0. We prove that the solutions always blow up in finite time for non‐trivial and non‐negative initial values. We also prove that the blow‐up occurs only on S_R = ?B_R for Ω = B_R = {x ? ?ⁿ:|x|<R}and under some assumptions on the initial values. Copyright © 2007 John Wiley & Sons, Ltd.     

Non-simultaneous blow up for coupled nonlinear diffusion equations with absorptions

This paper deals with coupled nonlinear diffusion equations with absorptions. We characterize the range of parameters for which non-simultaneous blow up occurs. We establish the necessary and sufficient conditions for the occurrence of non-simultaneous blow up with proper initial data. Moreover, we obtain the optimal condition under which any blow up is non-simultaneous.     

On the generalized exact boundary synchronization for a coupled system of wave equations

In this paper, we deliver a normalized synchronization transformation to study the generalized exact boundary synchronization for a coupled system of wave equations with Dirichlet boundary controls. The clear relationship among the generalized exact boundary synchronization, the exact boundary null controllability, and the generalized exactly synchronizable states is precisely obtained. This approach gives further a forthright decomposition for the generalized exact boundary synchronization problem, whereby, we gain directly the determination of generalized exactly synchronizable states.     

Partial approximate boundary synchronization for a coupled system of wave equations with Dirichlet boundary controls

In this paper, we propose the concept of partial approximate boundary synchronization for a coupled system of wave equations with Dirichlet boundary controls, and make a deep discussion on it. We analyze the relation-ship between the partial approximate boundary synchronization and the partial exact boundary synchronization, and obtain sufficient conditions to realize the partial approximate boundary synchronization and necessary conditions of Kalman's criterion. In addition, with the help of partial synchronization decomposition, a condition that the approximately synchronizable state does not depend on the sequence of boundary controls is also given.     

On a coupled system of fractional differential equations with coupled nonlocal and integral boundary conditions

We investigate a coupled system of fractional differential equations with nonlinearities depending on the unknown functions as well as their lower order fractional derivatives supplemented with coupled nonlocal and integral boundary conditions. We emphasize that the problem considered in the present setting is new and provides further insight into the study of nonlocal nonlinear coupled boundary value problems. We present two results in this paper: the first one dealing with the uniqueness of solutions for the given problem is established by applying contraction mapping principle, while the second one concerning the existence of solutions is obtained via Leray–Schauder’s alternative. The main results are well illustrated with the aid of examples.     

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.     

Induced generalized exact boundary synchronizations for a coupled system of wave equations

Taking a coupled system of wave equations with Dirichlet boundary controls as an example,by splitting and merging some synchronization groups of the state variables cor-responding to a given generalized synchronization matrix,this paper introduces two kinds of induced generalized exact boundary synchronizations to better determine its generalized exactly synchronizable states.