Quenching profile for a quasilinear parabolic equation
Jong‐Shenq Guo, Bei Hu · Quarterly of Applied Mathematics · 2000
Introduction.We consider the following first initial boundary value problem:ut -(u")xx ~ x G (-/,/)> t > 0, (1.1) u(±l,t) = 1, t>0, (1.2)where 00, Z>0, and uq(x) >0, Vx G [-1, Z].Without loss of generality, we may assume that uq(x) is smooth and bounded above by 1 such that uo(±Z) = 1.Since uo(x) is positive, the local (in time) existence and uniqueness of a classical solution of the problem (1.1)-(1.3)are trivial (see [8]).Many results in quenching, such as single point quenching and profiles, are similar to those blow-up results ([3], [5] and the references therein).The system (1.1)-(1.3) in the case a c, tn -> T, and u(xn,tn) -> 0 as n -> oo.It is shown in [8] that there can only be finitely many quenching points that stay a positive distance away from the boundary |x| = I for any positive initial data.The purpose of this paper is to study how the solution tends to zero.For simplicity we shall only consider the symmetric case, i.e., the case that uq is symmetric with respect to x = 0 and is monotone increasing in \x\.It has been shown that (0, T) is the only possible