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Blow-Up Analysis for a Quasilinear Degenerate Parabolic Equation with Strongly Nonlinear Source
We investigate the blow-up properties of the positive solution of the Cauchy problem for a quasilinear degenerate parabolic equation with strongly nonlinear source , where , , and , ,, and give a secondary critical exponent on the decay asymptotic behavior of an initial value at infinity for the existence and nonexistence of global solutions of the Cauchy problem. Moreover, under some suitable conditions we prove single-point blow-up for a large class of radial decreasing solutions.
In this paper, we consider the following Cauchy problem to a quasilinear degenerate parabolic equation with strongly nonlinear source where , , ??,???,??, and the initial data is nonnegative bounded and continuous.
Equation (1.1) has been suggested as a mathematical model for a variety of physical problems (see ). 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 ).
One of the particular features of problem (1.1) is that the equation is degenerate at points where or . 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 defined in is called a weak solution of the Cauchy problem (1.1) if for every bounded open set with smooth boundary , , , and for all and all test functions . Moreover, for any .
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 of (1.1) when the initial data has slow decay near . For instance, in the following case we investigate the existence of global and nonglobal solutions for the Cauchy problem (1.1) in terms of and . 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  obtained the secondary critical exponent for the following porous medium type equation in high dimensions: where , or , is nonnegative bounded and continuous, and proved that for , there exists a secondary critical exponent such that the solution of (1.5) blows up in finite time for the initial data , which behaves like at if , and there exists a global solution for the initial data , which behaves like at if . Here, we say that the solution blows up in finite time; it means that there exists such that for all , but .
Mu et al.  studied the secondary critical exponent for the following -Laplacian equation with slow decay initial values: where , , and showed that, for , there exists a secondary critical exponent such that the solution of (1.6) blows up in finite time for the initial data which behaves like at if , and there exists a global solution for the initial data , which behaves like at if .
Recently, Mu et al.  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 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 is called a blow-up point if there exists a sequence such that , and as , where 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 , and , the Cauchy problem (1.1) has been investigated by Weissler in , and the author obtained that the solution blows up only at a single point. Galaktionov and Posashkov  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 and . Recently, when and , Mu and Zeng  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 be the space of all bounded continuous functions in . For , we define Moreover, we denote Our main results of this paper are stated as follows.
Theorem 1.3. For , , , , and , suppose that for some ; then the solution of the Cauchy problem (1.1) blows up in finite time.
Theorem 1.4. For , , , , and , suppose that for some and for some ; then there is such that the solution of the Cauchy problem (1.1) exists globally for all , and if , one has where , .
Remark 1.5. When , and , we have and .
Remark 1.6. It follows from Theorems 1.3 and 1.4 that the number gives another cut-off between the blow-up case and the global existence case. Therefore, the number is a new secondary critical exponent of the Cauchy problem (1.1). Unfortunately, in the critical case , we do not know whether the solution of (1.1) exists globally or blows up in finite time.
Remark 1.8. In , Afanas’eva and Tedeev also established the Fujita type results for (1.1) with . In particular, if , , they obtained that if , then every nontrivial solution blows up in finite time, and if , then the solution exists globally for a small initial data . We note that when in (1.1), if and , then and , while if and , then . Therefore, the results of Theorems 1.3 and 1.4 coincide with those in .
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 in a small neighborhood of the point , where , . We assume that the initial data satisfies the following condition: for , , and , , and for , , .
Theorem 1.9. Let , and let condition (H) hold. In addition, assume that the initial function satisfies Let be the blow-up time; then one has where that is, there is single-point blow-up at point .
Remark 1.10. By (1.11), the best upper estimate (1.12) obtained by our method has the following form: in . But, we do not give the lower bound estimate of the radial solution in a small neighborhood of the point , where , .
Remark 1.11. When or , the results of Theorem 1.9 are consistent with those in [24, 25], respectively. For , in , 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  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.
2. 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 is the solution of the Cauchy problem (1.1) and is the blow-up time. Let
where , Then, and
Using Young's inequality, we have
Since , it follows from (2.3) and (2.4) that
By , , and Hölder’s inequality, we obtain
Therefore, by (2.5) and (2.6), we have
Applying Jensen’s inequality, we obtain
Thus, it follows from (2.7) and (2.8) that
as long as
Hence, if satisfies
then increases and remains below for all .
And by (2.9) we have Therefore, from (2.11) and (2.12), we obtain that blows up in finite time : and get an estimate on the blow-up time of the solution as follows: Finally, it remains to verify the blow-up condition (2.11). Since for some , there exist two positive constants and such that for all , and we have By the definition of , , we can choose so small such that (2.11) holds. The proof of Theorem 1.3 is complete.
3. 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 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 ). By symmetric properties of (3.1), the solution is given by the following form where the positive function is the solution of the problem We shall prove the existence of solution to (3.4) by the following ordinary differential equation, and furthermore we obtain the nonincreasing property of the solution .
Firstly, given a fixed , we consider the following Cauchy problem: According to the standard of the Cauchy problem for ODE and the methods used in [7, 30], we can obtain that the solution of the Cauchy problem (3.5) is positive, and as ; moreover, for some .
Secondly, we shall prove that there exists a one-to-one correspondence between and . Indeed, this can be seen from the following relation: where is the solution of (3.5) for . Then, Therefore, we can deduce that, for each , there exists a positive, bounded, and global solution satisfying (3.4).
Finally, we shall prove that the solution is non-increasing, that is, is also non-increasing. To do this, we need the following lemmas.
Lemma 3.1. Let be the solution of (3.5); then
Lemma 3.2. If there exists such that , then for all .
Proof. We shall prove by contradiction. Assuming that Lemma 3.2 does not hold, it is easy to see that there exists such that Multiplying (3.5) by and integrating over with , we obtain It follows from (3.12) and (3.13) that equivalently, Letting in (3.15), we obtain the inequality , which is a contradiction. The proof of Lemma 3.2 is complete.
Lemma 3.3. The solution of (3.5) is monotone nonincreasing in .
Proof. Our method is based on the contradiction argument. Suppose that, for some , , by Lemma 3.1, there exists such that By Lemma 3.2, we have . Using the similar argument in Lemma 3.1, we obtain which is a contradiction with (3.14). The proof of Lemma 3.3 is complete.
Next, we apply the monotone properties of 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 , there exists a constant such that Taking and the self-similar solution of (3.1) defined as (3.3), since , there exists a positive constant such that Setting , it is easy to verify that for all , where and .
Let ; then is the solution of the following problem Taking and noting that is non-increasing, we have
Step 2. Set , where and are solutions of the following problem: By a direct calculation, we obtain that satisfies
Step??3. We shall prove that there exists a positive constant such that the problem (3.22) has a global solution with bounded in if . According to the standard theory of ODE, the local existence and uniqueness of solution of (3.22) hold. By (3.22), we have for ; furthermore, the solution is continuous as long as the solution exists and is finite.
From (3.22), when exists in , then is uniquely defined by Since and is increasing, we obtain By (3.22), (3.25), and , it follows that Let be a positive constant defined by Then from (3.26), and , we have for any , as long as exists globally.
On the other hand, by (3.22) and (3.25), we have Therefore, is also global.
Step??4. For any , where is defined as (3.27), the solution of (1.1) with initial value exists globally, and in . Moreover, there exists a positive constant such that The proof of Theorem 1.4 is complete.
4. Single Point Blow-Up
In this section, under some suitable assumptions, we shall prove that the blow-up set consists of the single point . Moreover, we also give the upper estimate of the solution in a small neighborhood of the point , where , .
First, we suppose that the solution is radially symmetric, that is, depending only on at a given time . Therefore, we study the following problem:
Proof of Theorem 1.9. It is based on the method in . The main idea is to apply the maximum principle to the auxiliary function , which is defined in (4.7), and to show that is small enough in . Then by integrating the obtained inequality and taking limit as , one can get upper bound of the solution . 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 , for all , such that Then, the weak solution is the limit function of the solution of the following regularized problem (see ):
By the standard methods used in [1, 32], the uniform estimates for the passage to the limit which do not depend on are established. Therefore, for any fixed , we may assume that, for all sufficiently large , the function satisfies the following conditions: where , do not depend on , and Moreover, by using condition (H) and the maximum principle in , we have
Step 2. Set where is given in (1.11).
By a direct calculation, we find that satisfies the following parabolic equation: where Since , it follows from (1.11) and (4.6) that .
Now we consider the coefficients and in . By (4.6) and , we have It follows from (4.4) that as , where , and we obtain the following estimate: where , are positive constants, which are independent of .
Therefore, we have the parabolic differential inequality where , satisfy (4.15).
Next, we consider the function on the parabolic boundary of . At first, it is easy to see that for all . By (1.10), we have , as for all . Finally, it follows from (1.11) that
Hence, for all sufficiently large , there exists on the parabolic boundary of and as .
In order to estimate in , we study the following ODE: which has the solution Taking the sequence such that it is obvious that where Setting we have on the parabolic boundary, and satisfies the following parabolic inequality: It follows from (4.15) that By the maximum principle (Chapter II, ), we obtain that in , that is,
Step 3. For large and , we have the following estimate where By (4.7) and (4.26), we have