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

Critical exponents for a non‐Newtonian polytropic filtration equation with weighted nonlocal inner sources

“…Therefore, we combine the auxiliary function method and the forward self‐similar solution to obtain the critical Fujita exponent of problems (1) and (2). Meanwhile, the second critical exponent is found by improving the techniques adopted in Ma and Fang …”

“…of the degenerate p-Laplacian equation with r > r c when K(x) ∈ L 1 (R N ) and K(x) ∼ |x| − for |x| large enough. In Ma and Fang, 30 it was found that the second critical exponent * = pq+(r−1)(N− ) q(r+s+1−p) with s > s c when K(x) = (1 + |x|) − and 0 ≤ < N, where r c and s c are given in Galaktionov and Levine 18 and Afanas'eva and Tedeev. 20 Meanwhile, they obtained the large time behavior of global solution and the life span of nonglobal solution.…”

Localization of Solutions to a Doubly Degenerate Parabolic Equation With a Strongly Nonlinear Source

“…Namely, when 1 < q ≤ q F , the solution of (1.1) always blows up in finite time, while for q > q F the blow-up occurs if u 0 is large enough but the blow-up does not occur if u 0 is small enough. On the other hand, when m = l = 1, Mu et al [17] gave a new critical exponent a * = p q−(p−1) for the quasilinear degenerate parabolic equation with slow decay initial values. Here, we say that the solution blows up in finite time, it means that there exists T ∈ (0, +∞) such that ||u(·, t)|| L ∞ < ∞ for all t ∈ [0, T ), but lim t→T − sup u(·, t) L ∞ = ∞.…”

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

“…For more reference on this topic, we refer the readers to see [1,5,6,24,28,32] and reference therein.…”

“…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).…”