On decay rate of quenching profile at space infinity for axisymmetric mean curvature flow

We study the motion of noncompact hypersurfaces moved by their mean curvature obtained by a rotation around x -axis of the graph a function y = u ( x, t ) (deﬁned for all x 2 R ). We are interested to estimate its proﬁle when the hypersurface closes open ends at the quenching (pinching) time T . We estimate its proﬁle at the quenching time from above and below. We in particular prove that u ( x, T ) » j x j ¡ a as j x j ! 1 if u ( x, 0) tends to its inﬁmum with algebraic rate j x j ¡ 2 a (as j x j ! 1 with a > 0).

but it does not provide the convergence rate.
We are interested in studying the profile of u(x, T (m)), especially the behavior as |x| → ∞ which is affected by initial data.
The equation for u is of the form supplemented by initial data The function u 0 is assumed to satisfy u 0 is bounded and uniformly continuous in R, The Cauchy problem (1)-(2) has a unique positive classical solution with the conditions (3)-(4) to the initial data (cf [4]). However, the solution quenches in finite time. For a given initial datum u 0 , we see Let v be a solution of (1) with initial datum m = inf x∈R u 0 (x). It is easily seen that and It is immediate that T (u 0 ) ≥ T (m) by a comparison argument. We treat the case T (u 0 ) = T (m). The notion of "minimal quenching time" was defined in [4], which is recalled below.
In [4] we characterized solutions of (1)- (2) quenching only at space infinity. The following conditions on initial data u 0 play essential roles in [4].
A. There exists a sequence {x k } ∈ R such that x k → ∞ and u 0 (x +x k ) → m a.e. in R as k → ∞.
For an initial datum satisfying (3)-(4), we proved in [4] the following results for the Cauchy problem (1)-(2): 2. For an initial datum satisfying u 0 ≡ m, the solution (1)-(2) quenches only at space infinity. For a solution u of (1)-(2) with minimal quenching time T (m), we call u(·, T (m)) the profile of u (at the quenching time T(m)). The hypersurface corresponding to u(·, T (m)) is called limit surface. These are related studies on blow-up at infinity for the reaction-diffusion equations [8,5,6,3,10,9,11] (see also [7]). We shall explain these papers at the end of this introduction. In particular, blow-up profile was discussed, for example, in [8] and [11] for a semilinear heat equation.

There exists a function u(·, T
In this paper we consider the relation between the profile of a quenching solution at quenching time T (m) and the form of initial data. Our goal, which is investigating the shape of limit surface, is similar to studying blowup profile. Inspired by the method used in [8, §2b] and [11, Theorems 1.3 and 1.5], we construct a subsolution and a supersolution of the form φ(T (m)−t+ g(x, t)) with some function g(x, t) decaying to zero at space infinity, where in order to estimate the profile at the quenching time. Let ψ(x) be a positive function satisfying the following conditions: √ ψ(x) is bounded and uniformly continuous in R; there exist constants C 1 > 0 and C 2 > 0 such that .
, we obtain algebraic decay at the space infinity. Corollary 1.4. Assume that there exist constants a 1 > 0, a 2 > 0, C I > 0 and C II > 0 such that Then We conclude this introduction by giving a short review on blow-up (or quenching) at the space infinity. Lacey [8] considered problems in a half line of u t = u xx + f (u) in R + = {x : x > 0} and constructed solutions blowing up only at space infinity. Gladkov [7] studied problems of the equation u t = u xx +f (x, t, u) in R + and showed that solutions of the problem uniformly converge as x → ∞ to the solution of the ODE obtained by dropping u xx in the equation.
Giga-Umeda [5] proved that blow-up only at space infinity occurs under the condition lim |x|→∞ u 0 (x) = sup x∈R u 0 (x) =: M and u 0 ≡ M for nonnegative solutions of u t = ∆u + u p in R n (cf. also [12] for a related study). For generalization, see [6] and a review article by Giga-Seki-Umeda [3]. More recently, Shimojō [11] discussed blow-up profile u(x, T ) := lim t→T u(x, t) for x ∈ R n . See also Seki-Suzuki-Umeda [10] and Seki [9] for quasilinear parabolic equations, which generalized the result of [6]. They also gave necessary and sufficient conditions for a solution to have "minimal blow-up time (or the least blow-up time)". See [9,10,3] for the precise definition of the last notion.

Profile at quenching
In order to prove Theorem 1.3, we construct a subsolution and supersolution of the form φ(T (m) − t + g(x, t)), as we have explained before. This is a modification of the method employed in [8] and [11] to study blow-up profile for a semilinear heat equation. The function

t)ψ(y)dy
with the Gauss kernel of heat equation is used there. However, because the problem which we treat here is a quasilinear equation, the Gauss kernel is not appropriate in our problem. We use the following function instead of G(x, t): where with α ≥ 0, β ≥ 0 and γ > 0 being constants. Note that this g γ α,β may be expressed by It is easily seen that the derivatives are calculated and estimated as follows: and Before proving the Theorem 1.3 we prepare two propositions.
We may choose with α 0 in (28), and then the constant C (or C ) depends only on C 1 , C 2 , a, T (m), C I (or C 1 , C 2 , a, T (m), C II ).