Stability and Instability of Group Invariant Asymptotic Profiles for Fast Diffusion Equations
This paper is concerned with group invariant solutions for fast diffusion equations in symmetric domains. First, it is proved that the group invariance of weak solutions is inherited from initial data. After briefly reviewing previous results on asymptoti
- PDF / 342,870 Bytes
- 16 Pages / 439.37 x 666.142 pts Page_size
- 72 Downloads / 174 Views
Abstract This paper is concerned with group invariant solutions for fast diffusion equations in symmetric domains. First, it is proved that the group invariance of weak solutions is inherited from initial data. After briefly reviewing previous results on asymptotic profiles of vanishing solutions and their stability, the notions of stability and instability of group invariant profiles are introduced under a similarly invariant class of perturbations, and moreover, some stability criteria are exhibited and applied to symmetric domain (e.g., annulus) cases. Keywords Fast diffusion equation · Asymptotic profile · Group invariance · Stability
1 Introduction In this paper, we are concerned with the Cauchy-Dirichlet problem for the fast diffusion equation, ∂t |u|m−2 u = Δu in Ω × (0, ∞), (1) u=0 u(·, 0) = u0
on ∂Ω × (0, ∞),
(2)
in Ω,
(3)
where Ω is a bounded domain of RN with smooth boundary ∂Ω, m > 2, ∂t = ∂/∂t, u0 ∈ H01 (Ω) and Δ stands for the N -dimensional Laplacian. By putting w = |u|m−2 u, Eq. (1) can be rewritten in a usual form of fast diffusion equation, ∂t w = Δ |w|r−2 w in Ω × (0, ∞) (4) with the exponent r = m/(m − 1) < 2. Fast diffusion equations arise in the studies of plasma physics (see [6]), kinetic theory of gases, solid state physics and so on. G. Akagi (B) Graduate School of System Informatics, Kobe University, 1-1 Rokkodai-cho, Nada-ku, Kobe 657-8501, Japan e-mail: [email protected] R. Magnanini et al. (eds.), Geometric Properties for Parabolic and Elliptic PDE’s, Springer INdAM Series 2, DOI 10.1007/978-88-470-2841-8_1, © Springer-Verlag Italia 2013
1
2
G. Akagi
It is well known that every solution u = u(x, t) of (1)–(3) vanishes at a finite time t∗ = t∗ (u0 ) ≥ 0 such that 1/(m−2) 1/(m−2) c(t∗ − t)+ ≤ u(·, t)H 1 (Ω) ≤ c−1 (t∗ − t)+ for all t ≥ 0 0
with some constant c > 0, provided that 2 < m ≤ 2∗ := 2N/(N − 2)+ (see [5, 7, 18]). Moreover, for the case that 2 < m < 2∗ , Berryman and Holland [7] studied asymptotic profiles φ(x) := lim (t∗ − t)−1/(m−2) u(x, t) tt∗
of solutions u = u(x, t) for (1)–(3). Now, let us address ourselves to the stability and instability of asymptotic profiles. Namely, our question is the following: For any initial data u0 ∈ H01 (Ω) sufficiently close to an asymptotic profile φ, does the asymptotic profile of the unique solution u = u(x, t) for (1)–(3) also coincide with φ or not? In [7] and [15], the stability of the unique positive asymptotic profile is discussed for nonnegative initial data in some special cases (e.g., N = 1). Recently, in [8], further detailed behaviors of nonnegative solutions near the extinction time are investigated. Moreover, in [3], the notions of stability and instability of asymptotic profiles are precisely defined for (possibly) sign-changing initial data, and furthermore, some criteria for the stability and instability are presented under 2 < m < 2∗ . Furthermore, they are applied to several concrete cases of the domain Ω (e.g., ball domains) and the exponent m. However, there are still cases (e.g., annular dom
Data Loading...
