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 andm 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. LetN ≥ 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


Introduction
In this paper, we consider the following Cauchy problem to a quasilinear degenerate parabolic equation with strongly nonlinear source 1.1 where N ≥ 1, p > 2, m , l , q > 1, and the initial data u 0 x is nonnegative bounded and continuous.Equation 1.1 has been suggested as a mathematical model for a variety of physical problems see 1 .For instance, it appears in the non-Newtonian fluids and is a nonlinear form of heat equation.Moreover, it can also be used to model the nonlinear heat propagation in a reaction medium see 2 .
One of the particular features of problem 1.1 is that the equation is degenerate at points where u 0 or ∇u 0. Hence, there is no classical solution in general and we introduce the following definition of weak solution see 3, 4 .
Definition 1.1.A nonnegative measurable function u x, t defined in R N × 0, T is called a weak solution of the Cauchy problem 1.1 if for every bounded open set Ω with smooth boundary ∂Ω, u ∈ C loc Ω × 0, T , u m , u l ∈ L p loc 0, T; W 1,p Ω , and for all 0 ≤ t 0 ≤ t ≤ T and all test functions ϕ ∈ C 1 0 Ω × 0, T .Moreover, lim Under some suitable assumptions, the existence, uniqueness and regularity of a weak solution to the Cauchy problem 1.1 and their variants have been extensively investigated by many authors see 5-7 and the references therein .
The first goal of this paper is to study the blow-up behavior of solution u x, t of 1.1 when the initial data u 0 x has slow decay near x ∞.For instance, in the following case u 0 x ∼ M|x| −a with M > 0, a ≥ 0, 1. 4 we investigate the existence of global and nonglobal solutions for the Cauchy problem 1.1 in terms of M and a.In recent years, many authors have studied the properties of solutions to the Cauchy problem 1.1 and their variants see 8-17 and the references therein .In particular, J.-S.Guo and Y. Y. Guo 18 obtained the secondary critical exponent for the following porous medium type equation in high dimensions: where p > 1, m > 1 or max{0, 1 − 2/N } < m < 1, u 0 x is nonnegative bounded and continuous, and proved that for p > p * m m 2/N , there exists a secondary critical exponent a * 2/ p − m such that the solution u x, t of 1.5 blows up in finite time for the initial data u 0 x , which behaves like |x| −a at x ∞ if a ∈ 0, a * , and there exists a global solution for the initial data u 0 x , which behaves like |x| −a at x ∞ if a ∈ a * , N .Here, we say that the solution blows up in finite time; it means that there exists where p > 2, q > 1, and showed that, for q > q * c p − 1 p/N , there exists a secondary critical exponent a * c p/ q 1 − p such that the solution u x, t of 1.6 blows up in finite time for the initial data u 0 x which behaves like |x| −a at x ∞ if a ∈ 0, a * c , and there exists a global solution for the initial data u 0 x , which behaves like |x| −a at x ∞ if a ∈ a * c , 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.
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 C b R N be the space of all bounded continuous functions in R N .For a ≥ 0, we define Moreover, we denote Our main results of this paper are stated as follows.
Theorem 1.3.For N ≥ 2, p > 2, m > 1, l > 1, and q > q * c l m p − 2 p/N , suppose that u 0 x ∈ Φ a for some a ∈ 0, a * c ; then the solution u x, t of the Cauchy problem 1.1 blows up in finite time.
Remark 1.6.It follows from Theorems 1.3 and 1.4 that the number a * c p/ q − l − m p − 2 gives another cut-off between the blow-up case and the global existence case.Therefore, the number a * c is a new secondary critical exponent of the Cauchy problem 1.1 .Unfortunately, in the critical case a a * c , 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 with m l.In particular, if u 0 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 u 0 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 u 0 x u 0 r satisfies the following condition:

and let condition (H) hold. In addition, assume that the initial function u
Let T be the blow-up time; then one has where that is, there is single-point blow-up at point x 0.
Remark 1.10.By 1.11 , the best upper estimate 1.12 obtained by our method has the following form: in R 1 × 0, T .But, we do not give the lower bound estimate of the radial solution u r, t in a small neighborhood of the point x, t , where x 0, t T .
Remark 1.11.When m l 1 or m l > 1, the results of Theorem 1.9 are consistent with those in 24, 25 , respectively.For 1 < q < l m p − 2 , in 29 , the authors obtained the results of global blow-up to arbitrary compactly supported initial data.
Remark 1.12.From Theorem 1.9, we obtain the same decay exponent as that of Theorem 1.2 in 29 by different methods.Moreover, it is interesting to see that the decay exponent of the upper estimate of Theorem 1.9 is also the same as the secondary critical exponent of Theorems 1.3 and 1.4.This paper is organized as follows.In Section 2, by using the energy method, we will obtain a blow-up condition and prove Theorem 1.3.In Section 3, using the comparison principle, we can construct a global supersolution to prove Theorem 1.4.Finally, we consider the single-point blow-up under some suitable conditions and prove Theorem 1.9 in Section 4.

Blow-Up Case
By using the energy method, we will obtain a blow-up condition corresponding to 1.1 .Therefore, we need to select a suitable test function as follows: Proof of Theorem 1.3.Suppose that u x, t is the solution of the Cauchy problem 1.1 and T is the blow-up time.Let

2.3
Using Young's inequality, we have

2.6
Therefore, by 2.5 and 2.6 , we have

2.7
Applying Jensen's inequality, we obtain Thus, it follows from 2.7 and 2.8 that as long as ∀t ∈ 0, T .

2.10
Hence, if E 0 satisfies 11 then E t increases and remains below C 0 for all t ∈ 0, T .And by 2.9 we have Therefore, from 2.11 and 2.12 , we obtain that u x, t blows up in finite time T 1/C 1 E q−1/s 0 : and get an estimate on the blow-up time T of the solution u x, t as follows:

2.13
Finally, it remains to verify the blow-up condition 2.11 .Since u 0 x ∈ Φ a for some a ∈ 0, a * c , there exist two positive constants M and R 0 > 1 such that u 0 x ≥ M|x| −a for all |x| ≥ R 0 , and we have y as e −|y| dy.

2.14
By the definition of A ε , 0 < a < a * c , we can choose 0 < ε ≤ 1/R 0 so small such that 2.11 holds.The proof of Theorem 1.3 is complete.

Global Existence
In this section, we shall prove Theorem 1.4 by constructing a global supersolution.To do this, we introduce the radially symmetric self-similar solution U M,a x, t to the following Cauchy problem: It is well known that the existence and uniqueness of the solution of 3.1 have been well established see 7 .By symmetric properties of 3.1 , the solution U M,a x, t is given by the following form where the positive function f M is the solution of the problem

3.4
We shall prove the existence of solution f M r to 3.4 by the following ordinary differential equation, and furthermore we obtain the nonincreasing property of the solution f M r .

3.5
According to the standard of the Cauchy problem for ODE and the methods used in 7, 30 , we can obtain that the solution g r of the Cauchy problem 3.5 is positive, and g r → 0 as r → ∞; moreover, lim r → ∞ r a g r M 3.6 for some M M η > 0.
Secondly, we shall prove that there exists a one-to-one correspondence between M ∈ 0, ∞ and η ∈ 0, ∞ .Indeed, this can be seen from the following relation: where g 1 r is the solution of 3.5 for η 1.Then, Therefore, we can deduce that, for each M > 0, there exists a positive, bounded, and global solution f M r satisfying 3.4 .Finally, we shall prove that the solution g r is non-increasing, that is, f M r is also non-increasing.To do this, we need the following lemmas.Proof.Integrating the 3.5 over 0, ε with ε > 0, we have Dividing by ε and taking ε → 0 in 3.10 , we obtain lim which implies that 3.9 holds.The proof of Lemma 3.1 is complete.
Proof.We shall prove by contradiction.Assuming that Lemma 3.2 does not hold, it is easy to see that there exists ε > 0 such that g r > 0, g r > 0 in r 0 , r 0 ε .

3.12
Multiplying 3.5 by r N−1 and integrating over r 0 , r with r ∈ r 0 , r 0 ε , we obtain r N−1 g m p−2 g l βr N g r r r 0 Nβr N−1 g r dr − r r 0 aβr N−1 g r dr.

3.13
It follows from 3.12 and 3.13 that

3.15
Letting r → r 0 in 3.15 , we obtain the inequality 1 ≤ 0, which is a contradiction.The proof of Lemma 3.2 is complete.
Proof.Our method is based on the contradiction argument.Suppose that, for some r 0 > 0, g r 0 > 0, by Lemma 3.1, there exists r 1 ∈ 0, r 0 such that g r 1 0 , g m p−2 g l r 1 ≥ 0.

3.16
By Lemma 3.2, we have g r 1 > 0. Using the similar argument in Lemma 3.1, we obtain lim which is a contradiction with 3.14 .The proof of Lemma 3.3 is complete.
Next, we apply the monotone properties of f M r to obtain the condition on the global existence of the solution to 1.1 .
Proof of Theorem 1.4.We prove Theorem 1.4 by the following steps.
Step 1.Since ϕ x ∈ Φ a , there exists a constant K > 0 such that 3.18 Taking M > K and the self-similar solution U M,a x, t of 3.1 defined as 3.3 , since lim r → ∞ r a f M r M > K, there exists a positive constant R 0 such that it is easy to verify that ϕ x ≤ U M,a x, t for all x ∈ R N , where t 0 ∈ 0, 1 and t λU M,a x, λ l m p−2 −1 t t 0 is the solution of the following problem

3.20
Taking η f M 0 and noting that f M r is non-increasing, we have

3.21
Step 2. Set v x, t A t w x, B t , where A t and B t are solutions of the following problem:

3.22
By a direct calculation, we obtain that v x, t satisfies

3.23
Step 3. We shall prove that there exists a positive constant λ 0 λ 0 ϕ such that the problem 3.22 has a global solution A t , B t with A t bounded in 0, T if λ ∈ 0, λ 0 .According to the standard theory of ODE, the local existence and uniqueness of solution A t , B t of 3.22 hold.By 3.22 , we have A t > 0, A t > 1 for t > 0; furthermore, the solution is continuous as long as the solution exists and A t is finite.From 3.22 , when A t exists in 0, t , then B t is uniquely defined by B t t 0 A l m p−2 −1 s ds.

3.24
Since p > 2 and A t is increasing, we obtain By 3.22 , 3.25 , and a > a

3.26
Let λ 0 λ 0 ϕ be a positive constant defined by
On the other hand, by 3.22 and 3.25 , we have Therefore, B t is also global.
Step 4. For any λ ∈ 0, λ 0 , where λ 0 λ 0 ϕ is defined as 3.27 , the solution u x, t of 1.1 with initial value u 0 x λϕ x exists globally, and u x, t ≤ v x, t in R N × 0, ∞ .Moreover, there exists a positive constant C such that 3.29 The proof of Theorem 1.4 is complete.

Single Point Blow-Up
In this section, under some suitable assumptions, we shall prove that the blow-up set consists of the single point x 0.Moreover, we also give the upper estimate of the solution u x, t in a small neighborhood of the point x, t , where x 0, t T .First, we suppose that the solution is radially symmetric, that is, depending only on r |x| at a given time t > 0. Therefore, we study the following problem:

4.1
Proof of Theorem 1.9.It is based on the method in 21 .The main idea is to apply the maximum principle to the auxiliary function ω i r, t , which is defined in 4.7 , and to show that ω i is small enough in 0, i × 0, T .Then by integrating the obtained inequality and taking limit as i → ∞, one can get upper bound of the solution u r, t .Therefore, we divide the proof into the following steps.
Step 1.Since problem 1.1 has no classical solution, we will construct the weak solution by means of regularization of the degenerate equation.Now define a strictly monotone sequence {ε i }, ε i > 0 for all i 1, 2, 3, . .., such that Then, the weak solution u r, t is the limit function of the solution of the following regularized problem see 31 :

4.3
By the standard methods used in 1, 32 , the uniform estimates for the passage to the limit which do not depend on ε i are established.Therefore, for any fixed ε i > 0, we may assume that, for all sufficiently large i, the function u i r, t satisfies the following conditions: where M 1 , M 2 do not depend on i, and u i r, t r | r 0 0 ∀t ∈ 0, T .

14
Abstract and Applied Analysis Moreover, by using condition H and the maximum principle in 33 , we have Step 2. Set where λ 0 is given in 1.11 .By a direct calculation, we find that w i r, t satisfies the following parabolic equation: where where M 3 , M 4 are positive constants, which are independent of i.Therefore, we have the parabolic differential inequality where c i , d i satisfy 4.15 .
Next, we consider the function w i r, t on the parabolic boundary of Q i,T .At first, it is easy to see that w i 0, t 0 for all t ∈ 0, T .By 1.10 , we have w i i, t ≤ λ 0 i N u q 0 i o 1 , as i → ∞ for all t ∈ 0, T .Finally, it follows from 1.11 that

4.17
Hence, for all sufficiently large i, there exists γ i sup w i > 0 on the parabolic boundary of Q i,T and γ i o 1 as i → ∞.
In order to estimate w i r, t in Q i,T , we study the following ODE: which has the solution Taking the sequence ε i such that  − λ 0 r.

Lemma 3 . 1 .
Let g r be the solution of 3.5 ; then lim | u i r | p−2 ≤ r 1−N α i δ i u q i

,
r 1 → 0, from 4.34 , we obtain the upper estimateu r 0 , t 0 ≤ q − l − m p − 2 p lm p−2 1/p−1 λ 1/p−1 0 − p−1 / q−l−m p−2 r − p/ q−l−m p−2 0 Mu et al. 19 studied the secondary critical exponent for the following p-Laplacian equation with slow decay initial values: ≤ 0 on the parabolic boundary, and z r, t satisfies the following parabolic inequality:z t ≤ a i z rr b i z r c i z − M 3 − c i w i d i − M 4 i N−1 ε ν i , in Q i,T .4.24It follows from 4.15 thatz t ≤ a i z rr b i z r c i z, in Q i,T .4.25By the maximum principle Chapter II, 33 , we obtain that z r, t ≤ 0 in Q i,T , that is,w i r, t ≤ w i t ≤ w i T α i in Q i,T .4.26Step 3.For large i and u i r, t r ∈ −M 2 , 0 , we have the following estimate