View source for Quenching

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

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110030/q1100301.png" /> be a positive constant, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110030/q1100302.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110030/q1100303.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110030/q1100304.png" /></td> </tr></table>

and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110030/q1100305.png" />. 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|First boundary value problem]])

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110030/q1100306.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110030/q1100307.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110030/q1100308.png" /> for some positive constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110030/q1100309.png" />. The solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110030/q11003010.png" /> is said to quench if there exists a finite time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110030/q11003011.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110030/q11003012.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110030/q11003013.png" /> is called the quenching time. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110030/q11003014.png" /> is positive, a necessary condition for (a1) to hold is:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110030/q11003015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

Under certain conditions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110030/q11003016.png" />, it was shown in [[#References|[a17]]] that (a2) implies (a1). Its multi-dimensional version was proved in [[#References|[a21]]] for a bounded convex domain with a smooth boundary, and extended in [[#References|[a10]]] to a more general forcing term in a piecewise-smooth bounded convex domain. Thus, in studying quenching, the necessary condition (such as in [[#References|[a1]]]) is also used.

Another direction in quenching is the study of the critical length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110030/q11003017.png" />, which is the length such that the solution exists globally for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110030/q11003018.png" />, and quenching (according to the necessary condition) occurs for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110030/q11003019.png" />. Existence of a unique critical length and a method for computing it were studied in [[#References|[a3]]], [[#References|[a18]]]; its multi-dimensional versions were studied in [[#References|[a1]]], [[#References|[a2]]], [[#References|[a10]]]. Results on whether quenching in infinite time is possible were recently extended to more general forcing terms in [[#References|[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 [[#References|[a14]]] that a quenching model gives a better approximation than the blow-up model.

What happens after quenching? It is described by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110030/q11003020.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110030/q11003021.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110030/q11003022.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110030/q11003023.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110030/q11003024.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110030/q11003025.png" />. This was studied in [[#References|[a9]]]. A multi-dimensional version was given in [[#References|[a22]]].

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

The above results were extended to degenerate parabolic equations in [[#References|[a12]]], [[#References|[a13]]], [[#References|[a15]]], [[#References|[a16]]]. For a coupled system, the blow-up of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110030/q11003026.png" /> and existence of a unique critical length were studied in [[#References|[a4]]], [[#References|[a6]]], [[#References|[a20]]]. Problems involving more general parabolic equations or different boundary conditions are given in [[#References|[a5]]], [[#References|[a7]]]. The concept of quenching has been extended to time-periodic solutions of weakly coupled parabolic systems in [[#References|[a8]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Acker,   B. Kawohl,   "Remarks on quenching"  ''Nonlinear Anal.'' , '''13'''  (1989)  pp. 53–61</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C.Y. Chan,   "Computation of the critical domain for quenching in an elliptic plate"  ''Neural Parallel Sci. Comput.'' , '''1'''  (1993)  pp. 153–162</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  C.Y. Chan,   C.S. Chen,   "A numerical method for semilinear singular parabolic quenching problems"  ''Quart. Appl. Math.'' , '''47'''  (1989)  pp. 45–57</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  C.Y. Chan,   S.S. Cobb,   "Critical lengths for semilinear singular parabolic mixed boundary-value problems"  ''Quart. Appl. Math.'' , '''49'''  (1991)  pp. 497–506</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  C.Y. Chan,   D.T. Fung,   "Quenching for coupled semilinear reaction-diffusion problems"  ''Nonlinear Anal.'' , '''21'''  (1993)  pp. 143–152</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  C.Y. Chan,   H.G. Kaper,   "Quenching for semilinear singular parabolic problems"  ''SIAM J. Math. Anal.'' , '''20'''  (1989)  pp. 558–566</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  C.Y. Chan,   L. Ke,   "Critical lengths for periodic solutions of semilinear parabolic systems"  ''Dynam. Systems Appl.'' , '''1'''  (1992)  pp. 3–11</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  C.Y. Chan,   L. Ke,   "Beyond quenching for singular reaction-diffusion problems"  ''Math. Methods Appl. Sci.'' , '''17'''  (1994)  pp. 1–9</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  C.Y. Chan,   L. Ke,   "Parabolic quenching for nonsmooth convex domains"  ''J. Math. Anal. Appl.'' , '''186'''  (1994)  pp. 52–65</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  C.Y. Chan,   L. Ke,   A.S. Vatsala,   "Impulsive quenching for reaction-diffusion equations"  ''Nonlinear Anal.'' , '''22'''  (1994)  pp. 1323–1328</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  C.Y. Chan,   P.C. Kong,   "Quenching for degenerate semilinear parabolic equations"  ''Applicable Anal.'' , '''54'''  (1994)  pp. 17–25</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  C.Y. Chan,   P.C. Kong,   "Solution profiles beyond quenching for degenerate reaction-diffusion problems"  ''Nonlinear Anal.'' , '''24'''  (1995)  pp. 1755–1763</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  C.Y. Chan,   P.C. Kong,   "A thermal explosion model"  ''Appl. Math. Comput.'' , '''71'''  (1995)  pp. 201–210</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  C.Y. Chan,   P.C. Kong,   "Channel flow of a viscous fluid in the boundary layer"  ''Quart. Appl. Math.'' , '''55'''  (1997)  pp. 51–56</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  C.Y. Chan,   P.C. Kong,   "Impulsive quenching for degenerate parabolic equations"  ''J. Math. Anal. Appl.'' , '''202'''  (1996)  pp. 450–464</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top">  C.Y. Chan,   M.K. Kwong,   "Quenching phenomena for singular nonlinear parabolic equations"  ''Nonlinear Anal.'' , '''12'''  (1988)  pp. 1377–1383</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top">  C.Y. Chan,   K.K. Nip,   "Quenching for coupled degenerate parabolic equations" , ''Nonlinear Problems in Applied Mathematics'' , SIAM  (1996)  pp. 76–85</TD></TR><TR><TD valign="top">[a21]</TD> <TD valign="top">  K. Deng,   H.A. Levine,   "On the blowup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q110/q110030/q11003027.png" /> at quenching"  ''Proc. Amer. Math. Soc.'' , '''106'''  (1989)  pp. 1049–1056</TD></TR><TR><TD valign="top">[a22]</TD> <TD valign="top">  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</TD></TR></table>