ABSTRACT
Then, the leading term in (102) is the first-order containing a′2 and then, for τ 1,
a2(τ) = −m∗τ + ... (|a′′2 | ∼ 1τ3 |a′2| ∼ 1τ2
) . (104)
For c1 = 0, the ODE a′′2 = −γ∗a22 + ... ,
with γ∗ > 0, gives another asymptotic
a(τ) = −m∗τ2 + ... , where m∗ = 6γ∗ . (105)
Consider the more plausible case (103). Then, similar to the previous analysis in the inner and outer region, we obtain, instead of (91) and (92), the following expansion in the intermediate layer (for convenience, here m∗ > 0):
v2(y, τ) = f∗ − m∗√2 y2
τ + ... = f∗ − C∗ (
)2 + ... ≡ f∗ − C∗ζ2 + ... ,
where ζ = y√ τ
and C∗ = m∗√2 . (106)
Matching this transition behavior with that in the outer region with the limit profile (96) for k = 2 yields the extra logarithmic factor that affects the behavior in both the inner (106) and outer regions I and II,
√ τ =
√| ln(1− t)|, so that ζ = x√ (1−t)| ln(1−t)| . (107)
and
Using (96) with k = 2 and the scaling (34), we have that the blow-up behavior of the original solution u2(x, t) in the outer region I is given by
u2(x, t) ∼ (1− t)− 2p−1 f∗ ( 1 +D∗ x
(1−t)| ln(1−t)| )− 2p−1 + ... as t → 1−, (108)
on compact subsets in ζ, where D∗ > 0 is a universal constant depending on p only. Finally, the expansion (108) in the outer region I penetrates into the last
outer region II that yields the final-time profile for this pattern u2(x, t). Then, instead of (99),
u2(x, 1 −) = A∗|x|− 4p−1
∣ ∣ ln |x|∣∣ 2p−1 + ... for small |x| > 0. (109)
Remark: similar patterns in reaction-diffusion problems. The idea of a center manifold behavior such as (108) with a logarithmically perturbed similarity variable as in (99) for semilinear parabolic equations goes back to Hocking, Stuartson, and Stuart’s paper [206] published in 1972. Countable sets of blow-up patterns on stable manifolds were also detected a long time ago for blow-up in the Frank-Kamenetskii equation (13) (see references in [393, 394]). Similar phenomena exist in blow-up for higher-order reactiondiffusion equations as in Section 2 [138]
ut = −uxxxx + |u|p−1u (p > 1).
