Blow-up in nonlinear heat equations

Abstract

In this paper we study the blowup problem of nonlinear heat equations. Our result show that for a certain family of initial conditions the solution will blowup in finite time, the blowup parameters satisfy some dynamics which are asymptotic stable, moreover we provide the remainder estimates. Compare to the previous works our approach is analogous to one used in bifurcation theory and our techniques can be regarded as a time-dependent version of the Lyapunov-Schmidt decomposition. * Supported by NSERC under Grant NA7901.This result was further improved in several papers (see e.g. [22,21,25,14,31,45,15,16,17,5,34,35,36]). A blowup solution satisfying the bound ( 5) is said to be of type I. This bound was proven under various by an interpolation result. Note that the first four eigenvectors of L 0 are e − αx 2 4 , xe − αx 2 4 , (αx 2 − 1)e − αx 2 4 and (αx 3 − 3x)e − αx 2 4 with the eigenvalues −2α, −α, 0 and α. Thus for the case n = 2, using that the integral kernel of e −rL0 is positive and therefore e −rL0 g ∞ ≤ f −1 g ∞ e −rL0 f ∞ for any f > 0 and using that e −rL0 e − α 4 z 2 = e 2αr e − α 4 z 2 and e −rL0 (αz 2 − 1)e − α 4 z 2 = (αz 2 − 1)e − α 4 z 2 , we find thatThis implies (85). To prove (86) we note that U 0 (x, y) is, by definition, the integral kernel of the operator e − α 4 z 2 e −rL0 e α 4 z 2 . Thus, taking g(x) = x n e − α 4 x 2 in (85) yields (86).A version of the following lemma is proved in [5].Lemma 12. For any function g and positive constants σ and r we have z −3 e αz 2 4 U α (σ + r, σ)P α g ∞ [e 2αr r(1 + r)β 1/2 (σ) + e −αr ] z −3 e αz 2 4 g ∞ .

Neck Pinching Dynamics under Mean Curvature Flow

“…In this section we prove lower and upper bounds for v defined in ( 11). The main tool we use is a generalized form of maximum principle from [27].…”

“…Here v is the same as in ( 11) and is related to ξ be ( 16) and ( 71), and we ordered the terms in F according to the leading power in y 2 .…”

“…In this section we re-parameterize the unknown function ξ(y, τ ) in such a way that the y 2 -term in the linear part of the new equation has a time-independent coefficient (cf [11]). …”

“…The case n = 3 of ( 112) was proved in [11], Proposition 10 (cf [9,16]). The proof of the other cases is similar, thus is omitted.…”

On blowup dynamics in the Keller–Segel model of chemotaxis

“…Hence we supplement equation (20) with three additional equations. Two of the equations can be chosen as in [29] to make the parameters a, b, and c satisfy a chosen relation, say f (a, b, c) = 0. In addition, we have the relations φ, ζ bci = 0, i = 0, 1,…”