Blow-up directions for quasilinear parabolic equations

Abstract

We consider the Cauchy problem for quasilinear parabolic equations ut = ∆φ(u) + f (u), with the bounded non-negative initial data u 0 (We study blow-up of non-negative solutions with the least blow-up time, i.e. the time coinciding with the blow-up time of a solution of the corresponding ordinary differential equation dv/dt = f (v) with the initial data u 0 L ∞ (R N ) > 0. Such a blow-up solution blows up at space infinity in some direction (directional blow-up) and this direction is called a blow-up direction. We give a sufficient condition on u 0 for directional blow-up. Moreover, we completely characterize blow-up directions by the profile of the initial data, which gives a sufficient and necessary condition on u 0 for blow-up with the least blow-up time, provided that f (ξ) grows more rapidly than φ(ξ).

Blow-Up at Space Infinity for Solutions of Cooperative Reaction-Diffusion Systems

“…Later Shimojo [32] also calculate the shape of blow-up profile uðx; TÞ :¼ lim t!T uðx; tÞ for x A R n precisely. See also Seki-Suzuki-Umeda [30] and Seki [29] for quasilinear parabolic equations, which generalize the result of [12].…”

Blow-Up at Space Infinity for Nonlinear Heat Equations

“…Moreover, the nonlinear term f can be taken from very wide class of functions. For example, f (u) = (1 + u){log(1 + u)} β with β > 2 is allowed in [14]. One of the author obtained the same results for a quasilinear parabolic equation which is a generalization of fast diffusion equation u t = ∆u m + f (u) with 0 < m < 1 in [13], although the assumption of f is a little stronger than [14].…”

“…Giga and Umeda [7] first showed two sufficient conditions on initial data for ψ ∈ S N −1 to be a blow-up direction or non-blow-up direction, respectively and proved that every direction satisfies each of the conditions by a supplementary argument. On the other hand, Seki, Suzuki and Umeda [14] established the formulation of Theorem 3.3(ii) via entirely different approach, adopting a regularizing argument (see Lemma 4.1.2). As above mentioned, their assumption on initial data is equivalent to that of Giga and Umeda [7].…”

“…Seki, Suzuki and Umeda [14] not only generalized the results of [7] to degenerate quasilinear parabolic equations u t = ∆φ(u) + f (u) but also gave necessary and sufficient conditions for a solution to blow up at minimal blow-up time and conditions for a direction to be a blow-up direction. The equation is a generalization of porus medium equation with reaction term; u t = ∆u m + f (u) with m ≥ 1.…”

“…2 Remark 4.0.1. Theorem 3.3 can be proved even for degenerate quasilinear parabolic equations of the form u t = ∆φ(u) + f (u) ( [14]). For that case, we employ the results on modulus of continuity due to DiBenedetto [3, Lemma 5.2] in order to get equicontinuity of {u n }.…”