Dynamical Systems by Alexander J.C.
By Alexander J.C.
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Example text
3; this is a fundamental property of scale-invariant equations. 4 explains the difference between strong and weak smoothing effects, an often ignored question which plays an important role in the sequel. Though we concentrate on equations with power nonlinearities, the practice of nonlinear diffusion leads to many filtration models where the nonlinearity is only approximately power-like. 6. 8 we briefly touch on the question of actual asymptotic behaviour, which deserves a text of its own. 9 treats the interesting limit m → ∞, so-called mesa problem.
26) Next, given any solution u(x, t) with data u 0 ∈ L 1 (Rn ), u 0 ≥ 0 and u 0 d x = M, we can choose K , L , T so that u˜ fulfils on top of the above properties the requirement of having L 1 -mass 1: u˜ 0 (x) d x = K u 0 (L x, 0) d x = K L −n M. We have two conditions, K m−1 L 2 = T and K M = L n , hence L = M (m−1)δ T δ , K = M −2δ T nδ 1 n n 30 Smoothing effect and time decay. Data in L (R ) or M(R ) with δ = 1/(n(m − 1) + 2) and free parameter T . But now, taking t = 1 and fixing the free parameter T > 0 at will we have u(T ) ∞ = 1 u(1) ˜ K ∞ = c ≤ c M 2δ T −nδ .
1 The source solution for the PME was introduced by Zeldovich and Kompanyeets [ZK50] in a particular case and completely analysed by Barenblatt [Ba52] in the period 1950–52. It was independently introduced in the West by Pattle in [Pa59] in 1958. Its role in describing the asymptotic behaviour will be commented upon below. 3 Thus, the ZKB family of point source solutions disappears at the critical exponent m = m c . 37). 2; but the role is not the same and maybe the closest relative in behaviour is the family of self-similar solutions of the second kind constructed in Chapter 7.