Blow-Up Analysis for a Quasilinear Degenerate Parabolic Equation with Strongly Nonlinear Source

and Applied Analysis 3 Mu et al. 19 studied the secondary critical exponent for the following p-Laplacian equation with slow decay initial values: ut div ( |∇u|p−2∇u ) u, x, t ∈ R × 0, T , u x, 0 u0 x , x ∈ R, 1.6 where p > 2, q > 1, and showed that, for q > q∗ c p − 1 p/N , there exists a secondary critical exponent ac p/ q 1 − p such that the solution u x, t of 1.6 blows up in finite time for the initial data u0 x which behaves like |x|−a at x ∞ if a ∈ 0, ac , and there exists a global solution for the initial data u0 x , which behaves like |x|−a at x ∞ if a ∈ ac,N . Recently, Mu et al. 20 also investigated the secondary critical exponent for the doubly degenerate parabolic equation with slow decay initial values and obtained similar results. On the other hand, in this paper, we will also consider single-point blow-up for the Cauchy problem 1.1 . It is interesting to study the set of blow-up points and the behavior of the solution u x, t at the blow-up point. In order to investigate single-point blow-up for the Cauchy problem 1.1 , we introduce the concept of the blow-up point. Definition 1.2. A point x ∈ Ω is called a blow-up point if there exists a sequence xn, tn such that xn → x, tn → T− and u xn, tn → ∞ as n → ∞, where T is blow-up time. In recent years, some authors also studied single-point blow-up for the Cauchy problem to nonlinear parabolic equations see 21, 22 and the references therein by different methods. In particular, when p 2, l 1 and N 1, the Cauchy problem 1.1 has been investigated by Weissler in 23 , and the author obtained that the solution blows up only at a single point. Galaktionov and Posashkov 24 studied the single-point blow-up and gave the upper and lower bound near the blow-up point for the Cauchy problem 1.1 when p > 2 and m l 1. Recently, when p > 2 and m l, Mu and Zeng 25 extended Galaktionov’s results to the doubly degenerate parabolic equation. For more works about single-point blow-up, we refer to 26, 27 , where the parabolic systems have been considered. Motivated by the above works, based on a modification of the energy methods, comparison principle, and regularization methods used in 15, 19, 21, 24 , we investigate the secondary critical exponent and single-point blow-up for the Cauchy problem 1.1 . Before stating the results of the secondary critical exponent, we start with some notations as follows. Let Cb R be the space of all bounded continuous functions in R . For a ≥ 0, we define Φ { φ x ∈ Cb ( R ) | φ x ≥ 0, lim |x|→∞ sup |x|φ x < ∞ } , Φa { φ x ∈ Cb ( R ) | φ x ≥ 0, lim |x|→∞ inf |x|φ x > 0 } . 1.7 Moreover, we denote q∗ c l m ( p − 2) p N , ac p q − l −m(p − 2) . 1.8 Our main results of this paper are stated as follows. 4 Abstract and Applied Analysis Theorem 1.3. For N ≥ 2, p > 2, m > 1, l > 1, and q > q∗ c l m p − 2 p/N , suppose that u0 x ∈ Φa for some a ∈ 0, ac ; then the solution u x, t of the Cauchy problem 1.1 blows up in finite time. Theorem 1.4. For N ≥ 2, p > 2, m > 1, l > 1, and q > q∗ c l m p − 2 p/N , suppose that u0 x λφ x for some λ > 0 and φ x ∈ Φ for some a ∈ ac,N ; then there is λ0 λ0 φ > 0 such that the solution u x, t of the Cauchy problem 1.1 exists globally for all t > 0, and if λ < λ0, one has ||u x, t ||∞ ≤ Ct−aβ, ∀t > 0, 1.9 where β 1/ a l m p − 2 − 1 p , C > 0. Remark 1.5. When p > 2, N ≥ 2 and q > q∗ c , we have q∗ c > 1 and 0 < ac < N. Remark 1.6. It follows from Theorems 1.3 and 1.4 that the number ac p/ q − l −m p − 2 gives another cut-off between the blow-up case and the global existence case. Therefore, the number ac is a new secondary critical exponent of the Cauchy problem 1.1 . Unfortunately, in the critical case a ac, we do not know whether the solution of 1.1 exists globally or blows up in finite time. Remark 1.7. When m l 1 or m l > 1, the results of Theorems 1.3 and 1.4 are consistent with those in 19, 20 , respectively. Remark 1.8. In 28 , Afanas’eva and Tedeev also established the Fujita type results for 1.1 withm l. In particular, if u0 x ∼ |x|−a, 0 < a < N, they obtained that if q < m p− 1 p/a , then every nontrivial solution blows up in finite time, and if q > m p − 1 p/a , then the solution exists globally for a small initial data u0 x . We note that when m l in 1.1 , if q > m p − 1 p/N and 0 < a < p/ q −m p − 1 , then 0 < a < N and q < m p − 1 p/a , while if q > m p − 1 p/N and p/ q − m p − 1 < a < N, then q > m p − 1 p/a . Therefore, the results of Theorems 1.3 and 1.4 coincide with those in 28 . Finally, we also consider single-point blow-up for a large number of radial decreasing solutions of the Cauchy problem 1.1 and give upper bound of the radial solution u r, t in a small neighborhood of the point x, t , where x 0, t T . We assume that the initial data u0 x u0 r satisfies the following condition: H u0 x u0 r ≥ 0 for r > 0, u0 0 > 0, and u0 r ∈ C1 R1 , u0 0 0, and u0 r ≤ 0 for r > 0, M0 sup u0 r < ∞, K0 sup |u0 r | < ∞. Theorem 1.9. Let N ≥ 1, p > 2, m > 1, l > 1, and q > l m p − 2 , and let condition (H) hold. In addition, assume that the initial function u0 x u0 r satisfies u q 0 r · r o 1 as r −→ ∞, 1.10 λ0 inf r>0, u0 r >0 ⎧ ⎪⎨ ⎪⎩ − ∣∣(um0 )′∣∣ p−2( u0 )′ ru q 0 ⎫ ⎪⎬ ⎪⎭ ∈ ( 0, p − 2 ( p − 2) N 1 2 ] . 1.11 Abstract and Applied Analysis 5 Let T be the blow-up time; then one has u r, t ≤ Cr−p/ q−l−m p−2 , r, t ∈ R1 × 0, T , 1.12and Applied Analysis 5 Let T be the blow-up time; then one has u r, t ≤ Cr−p/ q−l−m p−2 , r, t ∈ R1 × 0, T , 1.12