Large time behavior and life span for a quasilinear parabolic equation with slowly decaying initial values

Life span and a new critical exponent for a degenerate parabolic equation

“…In order to obtain a blow-up condition corresponding to (1.4), we have to modify the function e − |x| 2 used in [10,18] to fit to the exterior domain E R ⊂ R N , and introduce following two lemmas: Lemma 3.1 (Mochizuki and Suzuki [17]). Set…”

“…In [12] Huang et al have obtained similar results for the Cauchy problem of semilinear system of equations u t = u + v p , v t = v + u q with pq > 1. Recently, Mukai et al [18] (for the case 1 < m < p) and Guo [9] (for the case (1 − 2 N ) + < m < 1) have studied the Cauchy problem (1.2). It is shown that for p > p * m = m + 2 N there is a secondary critical exponent a * = 2 p−m such that the solution of (1.2) blows up in finite time for any initial value u 0 (x) which behaves like |x| −a at |x| = ∞ if a ∈ (0, a * ); and there are global solutions for initial value u 0 (x) which behaves like |x| −a at |x| = ∞ if a ∈ (a * , N).…”

Absence of global solutions to a class of nonlinear parabolic inequalities

“…In [18] Mukai et al presented some properties of solutions to (4.2) where initial data slowly decay near x = ∞. For instance, in the case u 0 (x) ∼ λ|x| −a the authors obtained global existence and nonglobal existence in terms of λ > 0 and a ≥ 0.…”

“…On the other hand, it is shown in [18] that if a < a < N where β = 2 and p > m + 2/N , then there exist global solutions to u t = ∆u m + u p such that the limit lim |x|→∞ u(x, 0)|x| a is finite and positive. Based on this observation it is natural to address the following question.…”

Задача Коши Для Вырождающегося Параболического Уравнения С Неоднородной Плотностью И Источником В Классе Медленно Стремящихся К Нулю Начальных Функций

“…Если начальная функция интегрируема лишь локально, то, как показано в [22], значение пока-зателя p * зависит от поведения начальной функции при |x| → ∞. В работе [22] …”