Asymptotic Behavior of Zeros of Solutions for Parabolic Equations

Abstract

Let p>1, : 0, and . # L (R) & L 1 (R) change its sign finite times. This paper is concerned with a Cauchy problemDefine the set of zeros of a solution u by Z(t)=[x # R : u(x, t)=0] for t>0. In the case of :=0, we show that the set Z(t) is contained in [&Ct, Ct] for large t>0 with some C>0 and that this order of t is best possible. When :>0, we also give estimates of Z(t) for global solutions and prove that Z(t)/[&K, K] for all t # (0, T ) with some K>0 for each blowup solution, where T is the blowup time.2001 Academic Press

Asymptotic profiles of zero points of solutions to the heat equation

“…) in the statement of Theorem 2.3 always holds by Theorem 1.1 in [4]. We thus deduce the following result from a proof by contradiction.…”

“…The analysis of the zero level set is important to understanding the behavior of signchanging solutions of parabolic equations. To analyze the behavior of sign-changing solutions, many researchers have focused on the zero level set or the sign-changing number of the initial data [1,2,3,4,5]. In the case of parabolic equations, Angenent [1] proved that Z(t) is a discrete subset of R. Furthermore, it is known that the number of elements of Z(t) does not increase as time passes [2,3].…”

“…In investigating the dynamics of the sign-changing solution in detail, the asymptotic behavior of Z(t) has been analyzed to the heat equation. Mizoguchi [4] estimated the upper bound of Z(t) in the case that f (x) is sign-changing at finite times and proved Z(t) ⊂ [−C 0 t, C 0 t] for sufficiently large t > 0 with some C 0 > 0. Moreover, it has been reported that there exist f (x) and x * (t) ∈ Z(t) such that x * (t) > C 1 t holds for sufficiently large t > 0 with some C 1 > 0.…”

Long-time asymptotics of the zero level set for the heat equation

“…Remark 3.2. For the one-dimensional heat equation, Mizoguchi [17] proved that any zero level set Z(t) is contained in [−Ct, Ct] for large t > 0 with some C > 0 if the initial data changes sign a finite number of times. Because u(x, t) := e G(x, t) − G(x − 2, t) has a unique zero z(t) = t + 1, Mizoguchi's upper bound t is optimal.…”