Quenching for a Diffusion System with Coupled Boundary Fluxes

In this paper, we investigate a diffusion system of two parabolic equations with more general singular coupled boundary fluxes. Within proper conditions, we prove that the finite quenching phenomenon happens to the system. And we also obtain that the quenching is non-simultaneous and the corresponding quenching rate of solutions. This extends the original work by previous authors for a heat system with coupled boundary fluxes subject to non-homogeneous Neumann boundary conditions.


Introduction
In the present work, we mainly deal with the following diffusion system with singular coupled boundary fluxes In the model (1), u and v can be thought as the temperatures of two mixed media during the heat propagation. This is a one-dimensional heat conduction rod of length 1 with positive initial temperatures 0 ( ) u x , 0 ( ) v x . At the left end {x=0}, heat is taken away with a rate ( (0, )) f v t and ( (0, )) g u t for u and v , respectively. The right end {x=1} Since the assumption proposed for the system implies that the two components are coupled completely and enhanced each other in the model. It is known that the singular negative flux at the boundary {x=0} may result in the so called finite time quenching of solutions, which makes it so interesting to investigate the quenching phenomenon of the solutions, see [2-5, 7-8, 11-13] and some survey papers [1,6,10]. Right here, we say that the solution ( , ) u v of the problem (1) quenches, if ( , ) u v exists in the classical sense and is positive for all 0 t T ≤ < and satisfies 0 1 If this happens, T will be called as quenching time. Since a singularity develops in the absorption term at quenching time T, thus the classical solution doesn't exist anymore.
Due to the great work by many previous researchers, the blow-up problems of parabolic equation have been studied gradually matured, thus plenty of authors have begun to pay attention to the quenching phenomena and become a heated study field.
Ferreira, Pablo and Quirs. etc in [2] studied a system of heat equations coupled at the boundary (2) They obtained that if , 1 p q ≥ , and then quenching is always simultaneous. While if p 1 < or q 1 < , non-simultaneous quenching indeed occurs. If 0 , 1 p q < < , then there exists initial data such that simultaneous quenching produces. Besides, if quenching is non-simultaneous and, for instance u is the quenching variable, then and (0, )ũ T x , where and throughout this paper, the notation f g means that there exist two positive constants for t close to the quenching time T .
They obtained the quenching rate is Ji, Qu and Wang in [5] considered finite time quenching problem for parabolic system 0 0 Zhi and Mu in [7] studied the non-simultaneous quenching in a semilinear parabolic system All of them have identified simultaneous and nonsimultaneous quenching by a precise classification of parameters, and establish simultaneous quenching rates or non-simultaneous quenching rates.
Motivated by those papers and references therein, the main purpose of this paper is to study a more general system (1) to obtain a lot of more general conclusions for the non-simultaneous quenching phenomenon with coupled fluxes at the boundary, which appeared in many papers with some special case, see [2][3][4][5][6][11][12][13]].

Main Results and Proof
In this section, we mainly deal with the non-simultaneous quenching, quenching rates and quenching set.
At first, we will prove a priori estimate to begin our study, which ensures that quenching always happens for the diffusion system (1.1). To simplify the presentation of the proofs, we define the functions as follows .
Lemma 2.1 Quenching happens for system (1.1) for every initial data.
Proof: By the maximum principle we have Therefore, by integrating 1 (1) in the interval [0, 1], we can with δ small sufficiently. Thus by the maximum principle, we can obtain that ( , ), ( , ) 0 F x t G x t ≤ for every The result in (2.5) is just the particular case for 0 x = .
Moreover, we have the following estimates via directly integrating for inequalities (2.5).
Within these estimates we can obtain the following corollary. Proof: In Lemma 2.1, we have given the lower bound of the non-simultaneous rate, while the upper bound can be obtained easily by integrating the first estimate in (11). Using that C>0 As 0 x → , by lower estimate given in Corollary 2.1, then upper estimate follows directly from the fact that u is concave; therefore To this end, the proof of Theorem 2.1 is complete.

Conclusion
Throughout this paper, we have studied the solutions of a parabolic system of heat equations coupled at the boundary through a singular flux. This system displays a singularity in finite time, which is called quenching in the literature. We obtained the quenching point is the origin, non-simultaneous quenching rates. To some degree, our work extends the original work by previous authors for a heat system with coupled boundary fluxes for a more general boundary flux.
We have to admit that there are still many possible improvements and extensions of our results. One possibility is that we consider the diffusion process in a higher dimension. If we study the radial solutions in a ball, some similar results may hold as well. Besides, we can extend the local diffusion to nonlocal diffusion, which may be more effective to describe the real situation. Another aspect for us to improve is to find a method to identify the non-simultaneous quenching and simultaneous quenching, which once was determined by some parameters.