Quenching

parabolic quenching, critical size and blow-up of the time-derivative

Let $ a $ be a positive constant, $ T \leq \infty $, $ \Omega = ( 0,a ) \times ( 0,T ) $,

$$ \partial \Omega = ( [ 0,a ] \times \{ 0 \} ) \cup ( \{ 0,a \} \times ( 0,T ) ) , $$

and $ Hu = u _ {xx } - u _ {t} $. The concept of quenching was introduced in 1975 through the study of a polarization phenomenon in ionic conductors. Consider the singular first initial-boundary value problem (cf. also First boundary value problem)

$$ Hu = - f ( u ) \textrm{ in } \Omega, $$

$$ u = 0 \textrm{ on } \partial \Omega, $$

where $ {\lim\limits } _ {u \rightarrow c ^ {-} } f ( u ) = \infty $ for some positive constant $ c $. The solution $ u $ is said to quench if there exists a finite time $ T $ such that

$$ \tag{a1 } \sup \left \{ {\left | {u _ {t} ( x,t ) } \right | } : {0 \leq x \leq a } \right \} \rightarrow \infty \textrm{ as } t \rightarrow T ^ {-} . $$

Here, $ T $ is called the quenching time. When $ u _ {t} $ is positive, a necessary condition for (a1) to hold is:

$$ \tag{a2 } \max \left \{ {u ( x,t ) } : {0 \leq x \leq a } \right \} \rightarrow c ^ {-} \textrm{ as } t \rightarrow T ^ {-} . $$

Under certain conditions on $ f $, it was shown in [a17] that (a2) implies (a1). Its multi-dimensional version was proved in [a21] for a bounded convex domain with a smooth boundary, and extended in [a10] to a more general forcing term in a piecewise-smooth bounded convex domain. Thus, in studying quenching, the necessary condition (such as in [a1]) is also used.

Another direction in quenching is the study of the critical length $ a ^ {*} $, which is the length such that the solution exists globally for $ a < a ^ {*} $, and quenching (according to the necessary condition) occurs for $ a > a ^ {*} $. Existence of a unique critical length and a method for computing it were studied in [a3], [a18]; its multi-dimensional versions were studied in [a1], [a2], [a10]. Results on whether quenching in infinite time is possible were recently extended to more general forcing terms in [a19]. This also answers the question what happens at the critical length (or size for the multi-dimensional version).

In a thermal explosion model using the Arrhenius law, it is shown in [a14] that a quenching model gives a better approximation than the blow-up model.

What happens after quenching? It is described by

$$ Hu = - f ( u ) \chi ( \{ u < c \} ) \textrm{ in } \Omega, $$

$$ u = 0 \textrm{ on } \partial \Omega, $$

where $ \chi ( s ) = 1 $ if $ u \in S $, and $ \chi ( S ) = 0 $ if $ u \notin S $. This was studied in [a9]. A multi-dimensional version was given in [a22].

The impulsive effect on quenching was first studied in [a11].

The above results were extended to degenerate parabolic equations in [a12], [a13], [a15], [a16]. For a coupled system, the blow-up of $ u _ {t} $ and existence of a unique critical length were studied in [a4], [a6], [a20]. Problems involving more general parabolic equations or different boundary conditions are given in [a5], [a7]. The concept of quenching has been extended to time-periodic solutions of weakly coupled parabolic systems in [a8].

References

[a1]	A. Acker, B. Kawohl, "Remarks on quenching" Nonlinear Anal. , 13 (1989) pp. 53–61
[a2]	C.Y. Chan, "Computation of the critical domain for quenching in an elliptic plate" Neural Parallel Sci. Comput. , 1 (1993) pp. 153–162
[a3]	C.Y. Chan, C.S. Chen, "A numerical method for semilinear singular parabolic quenching problems" Quart. Appl. Math. , 47 (1989) pp. 45–57
[a4]	C.Y. Chan, C.S. Chen, "Critical lengths for global existence of solutions for coupled semilinear singular parabolic problems" Quart. Appl. Math. , 47 (1989) pp. 661–671
[a5]	C.Y. Chan, S.S. Cobb, "Critical lengths for semilinear singular parabolic mixed boundary-value problems" Quart. Appl. Math. , 49 (1991) pp. 497–506
[a6]	C.Y. Chan, D.T. Fung, "Quenching for coupled semilinear reaction-diffusion problems" Nonlinear Anal. , 21 (1993) pp. 143–152
[a7]	C.Y. Chan, H.G. Kaper, "Quenching for semilinear singular parabolic problems" SIAM J. Math. Anal. , 20 (1989) pp. 558–566
[a8]	C.Y. Chan, L. Ke, "Critical lengths for periodic solutions of semilinear parabolic systems" Dynam. Systems Appl. , 1 (1992) pp. 3–11
[a9]	C.Y. Chan, L. Ke, "Beyond quenching for singular reaction-diffusion problems" Math. Methods Appl. Sci. , 17 (1994) pp. 1–9
[a10]	C.Y. Chan, L. Ke, "Parabolic quenching for nonsmooth convex domains" J. Math. Anal. Appl. , 186 (1994) pp. 52–65
[a11]	C.Y. Chan, L. Ke, A.S. Vatsala, "Impulsive quenching for reaction-diffusion equations" Nonlinear Anal. , 22 (1994) pp. 1323–1328
[a12]	C.Y. Chan, P.C. Kong, "Quenching for degenerate semilinear parabolic equations" Applicable Anal. , 54 (1994) pp. 17–25
[a13]	C.Y. Chan, P.C. Kong, "Solution profiles beyond quenching for degenerate reaction-diffusion problems" Nonlinear Anal. , 24 (1995) pp. 1755–1763
[a14]	C.Y. Chan, P.C. Kong, "A thermal explosion model" Appl. Math. Comput. , 71 (1995) pp. 201–210
[a15]	C.Y. Chan, P.C. Kong, "Channel flow of a viscous fluid in the boundary layer" Quart. Appl. Math. , 55 (1997) pp. 51–56
[a16]	C.Y. Chan, P.C. Kong, "Impulsive quenching for degenerate parabolic equations" J. Math. Anal. Appl. , 202 (1996) pp. 450–464
[a17]	C.Y. Chan, M.K. Kwong, "Quenching phenomena for singular nonlinear parabolic equations" Nonlinear Anal. , 12 (1988) pp. 1377–1383
[a18]	C.Y. Chan, M.K. Kwong, "Existence results of steady-states of semilinear reaction-diffusion equations and their applications" J. Diff. Eq. , 77 (1989) pp. 304–321
[a19]	C.Y. Chan, H.T. Liu, "Quenching in infinite time on the N-dimensional ball" Dynamics of Continuous, Discrete and Impulsive Systems (An Internat. J. for Theory and Applications) (to appear)
[a20]	C.Y. Chan, K.K. Nip, "Quenching for coupled degenerate parabolic equations" , Nonlinear Problems in Applied Mathematics , SIAM (1996) pp. 76–85
[a21]	K. Deng, H.A. Levine, "On the blowup of $u_t$ at quenching" Proc. Amer. Math. Soc. , 106 (1989) pp. 1049–1056
[a22]	M. Fila, H.A. Levine, J.L. Vazquez, "Stabilization of solutions of weakly singular quenching problems" Proc. Amer. Math. Soc. , 119 (1993) pp. 555–559