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

Abstract

In this paper, we consider the positive solution of the Cauchy problem for the equationand give a secondary critical exponent of the behavior of initial value at infinity for the existence of global and nonglobal solutions of the Cauchy problem. Furthermore, the life span of solutions are also studied.

Life span and a new critical exponent for a quasilinear degenerate parabolic equation with slow decay initial values

“…In order to obtain a blow-up condition corresponding to (1.5), we have to modify the function e −ε|x| 2 used in [14,19,23], and introduce a new test function φ ε (x) as follows.…”

“…To this, we start with the following lemma used to obtain a lower estimate of T * λ . [19,23]). For q > 1.…”

“…Motivated by their works, in [19] we have considered the Cauchy problem (1.4) for the case of N ≥ 2, p > 1 and q > p + 1 + 2 N , and proved that there is a new secondary critical exponent a * = 2 q−p−1 such that the solution of (1.4) 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).…”

Blow-up phenomena for polytropic equation with inhomogeneous density and source

“…2,9,17,and 26). The critical exponents for the one-dimensional equation with nonlinear boundary sources and ρ (x) = 1 were considered in Refs.…”

Secondary critical exponent, large time behavior and life span for a quasilinear parabolic equation with slowly decaying initial values

“…In [26], Li and Mu have considered the Cauchy problem (1.4) for the case of N ≥ 2, p > 1, and q > p + 1 + 2 N and proved that there is a new secondary critical exponent a * = 2 q−p−1 such that the solution (1.4) blows up in finite time for any initial value u 0 (x), which behave like |x| −a and |x| = ∞ if a ∈ (0, a * ); and there are global solutions for the initial value u 0 (x), which behaves like |x| −a at |x| = ∞ if a ∈ (a * , N). Then in [28], Mu, Li, and Wang have studied problem (1.5) for the case N ≥ 2, p > 2, and q > p − 1 + p N and also proved there is a new secondary critical exponent a * = p q+1−p such that the solution (1.5) blows up in finite time for any initial value u 0 (x), which behave like |x| −a and |x| = ∞ if a ∈ (0, a * ); and there are global solutions for the initial value u 0 (x), which behaves like |x| −a at |x| = ∞ if a ∈ (a * , N).…”