Abstract
This work is concerned with the global existence and nonexistence of solutions for a quasilinear parabolic equation with null Dirichlet boundary condition. Based on the Galerkin approximation technique and the theory of a family of potential wells, we obtain the invariant sets and vacuum isolating of global solutions including critical case, and we also give global nonexistence.
MSC: 35A01, 35B06, 35B08.
Keywords:
family of potential wells; global existence; nonexistence; vacuum isolating; critical value1 Introduction
Our main interest lies in the following quasilinear pLaplacian parabolic equation:
subject to homogeneous Dirichlet boundary and initial conditions
where , () is a bounded domain with smooth boundary, if ; if and . For simplicity, we denote by and .
Many natural phenomena have been formulated as the nonlinear diffusive equation (1.1) such as the model of nonNewton flux in the mechanics of a fluid, the model of a population, biological species and filtration; we refer to [1,2] and the references therein. In the nonNewtonian theory, the quantity p is a characteristic of the medium. Media with are called dilatant fluids, while media with are called pseudoplastics. If , they are Newtonian fluids.
There have been a lot of results on the global existence and nonexistence of solutions for nonlinear evolution equations and many effective methods have been developed such as the compactness method, the semigroup method, continuation of local solutions, the upperlower solution method, and the concavity method; see [36]. Since the potential well was introduced by Sattinger [7] in order to prove the global existence of solutions for nonlinear hyperbolic equations which do not necessarily have positive definite energy, it has become an important method to study the global existence and nonexistence of solutions for various nonlinear evolution equations. It is an effective method to get a positive definite energy through one term, controlled by the other term in the principal part of the equations [719]. For instance, Levine [8] investigated the initial boundary value problem of the linear heat conduction equation
with nonlinear boundary condition, and he obtained the global existence and nonexistence of weak solutions. Payne and Sattinger [9] investigated the initial boundary value problem of the semilinear hyperbolic equation with fully nonlinear term
and proved the weak solution blows up in finite time. They also extended the results to the initial boundary value problem of the corresponding parabolic equation. Tsutsumi [10] studied the homogeneous Dirichlet initial boundary value problem of the nonlinear parabolic equation
and he obtained the sufficient conditions of the existence of global weak solutions and the solutions blow up in finite time for the case . Later, Liu [11] proved the global existence of solutions of the homogeneous Dirichlet initial boundary value problem for (1.4) with critical initial conditions. Pang and Zhang [12] investigated the initial boundary value problem of the quasilinear parabolic equation
and they obtained the sufficient conditions as regards global existence and nonexistence of solution by using a potential well method.
However, the potential wells used in these works were defined by the same method as Sattinger [7] and their results were similar. Until Liu [20] firstly introduced the theory of a family of potential wells, described the structure of potential wells and the estimates of the depth of potential wells. And he firstly found the phenomenon of vacuum isolating of solutions for nonlinear evolution equations. The study of applications about a family of potential wells has attracted more and more attention [2025]. For instance, Liu and Zhao [21] not only proved the global existence and nonexistence of solutions, but they also obtained the vacuum isolating of solutions of the initial boundary value problem for semilinear hyperbolic equations and parabolic equations.
As far as we know, there are fewer papers on the global existence and nonexistence of weak solutions for nonlinear parabolic equations by using the theory of a family of potential wells. In particular, for our problem (1.1)(1.3), the analysis of the structure and depth of the potential well, the invariant sets, the vacuum isolating of global solutions, and the question of the global existence of solutions with critical initial conditions are still open. It is difficult to obtain an a priori estimate of the approximate solution for the study of the existence of global solutions by using the general Galerkin approximation method, but the theory of the potential well often makes up for the defect. The combination of the two methods can be used to solve the existence of solutions effectively. Moreover, the study of the phenomenon of vacuum isolating will be helpful for us in studying the distribution of solutions in Sobolev space. But the depth of potential well d for the problem (1.1)(1.3) is usually very small etc. Our goal is to improve the theory of a family of potential wells for studying the global existence and nonexistence of solutions for our problem (1.1)(1.3), including the critical case, and we further generalize the results in [1012,20].
The outline of the paper is as follows. In Section 2, we firstly give the definition of the weak solution for problem (1.1)(1.3), and the definition and properties of a family of potential wells. Then we prove the global existence of solutions for problem (1.1)(1.3) by using the Galerkin approximation technique and the theory of a family of potential wells in Section 3. The invariant sets of global solutions and vacuum isolating are obtained in Section 4. Then the sufficient condition of global nonexistence of solutions is given in Section 5. Finally, we give the result of global existence with critical initial conditions.
2 Preliminaries
Due to the degeneracy of (1.1), problem (1.1)(1.3) has no classical solutions in general. We need to give the definition of the weak solution firstly.
Definition 1 A function is called a weak solution of problem (1.1)(1.3) on if it satisfies the following conditions:
where T is either infinity or the limit of the existence interval of solution.
In order to study the problem (1.1)(1.3), we also consider the auxiliary equation
Next, we define the functionals , , and the potential well W as follows:
Furthermore, for problem (1.1)(1.3) and , we define the auxiliary functional and function
In the paper, we always assume that p and q satisfy (H):
Before giving our main results, we show some preliminary lemmas which are very important in the following proofs. As for the proofs of these several lemmas, we will not repeat them again (see [11,20]).
Lemma 1 ([11], Lemma 2.2)
For any given, , possesses the following properties:
(2) There exists a uniquesuch that;
(3) , i.e. is increasing for; , i.e. is decreasing for;
Lemma 2 ([20], Lemmas 2.12.3)
The following sufficient and necessary conditions always hold:
Lemma 3 ([20], Lemma 2.4)
The functionpossesses the following properties on the interval:
(2) takes the maximumat, where, ;
(3) is increasing onand decreasing on;
(4) For any given, the equationhas exactly two rootsand.
Lemma 4 ([20], Lemma 2.5)
Proof The result can easily be obtained by Lemma 4 and the fact that is equivalent to . □
Now we can define a family of potential wells as follows:
Remark 1 From , we see that implies that .
In the following, we define
Note that , hence for any given , when , we have and . This implies that , where satisfies .
Lemma 5 ([20], Theorem 2.7)
Suppose that, , , , andare defined as the above, then
Lemma 6 ([20], Lemma 2.10)
Assume thatfor some given, are the two roots of the equation, then the sign ofis not changed for.
Lemma 7 ([11], Lemma 2.8)
Letpandqsatisfy (H), then the solutions given in Theorem 1 satisfy
3 Existence of global weak solutions
In this section, we obtain the global existence of solutions for problem (1.1)(1.3) by combining the Galerkin approximation technique and the theory of a family of potential wells.
Theorem 1Assume thatpandqsatisfy (H), . If, are the two roots of the equationandor, then problem (1.1a)(1.3) admits a global weak solutionsuch that, andforand. Furthermore, we have
(2) If, then the solution is uniquely determined by the initial function;
(3) Ifa.e. in Ω, the solutiona.e. in Ω for any fixed, henceis a solution of the problem (1.1)(1.3).
Proof Let be a system of base functions of . Construct approximate solutions in the form
satisfying
Multiplying (1.1a) by , summing over s and integrating with respect to t, we obtain
Note that implies . By Lemma 6, we have for . From this and , we obtain for . If , then for . For any fixed , we have and (if ) or (if and is defined in Lemma 5), thereby for sufficiently large n.
Next, we prove that for sufficiently large n and . Otherwise, there must be a such that , i.e. and or . From (3.1), we have
hence is impossible. If and , then by Lemma 4, we have , which is also impossible. Thus from (3.1) and Lemma 2, we obtain
and
for and sufficiently large n. From these and the compactness method, we can prove that problem (1.1a)(1.3) admits a global weak solution such that , and for any and .
Furthermore, by Theorem 1 in [10] we can easily get the results (1)(3), here we omit the proofs. □
Similarly, we can get the following conclusions directly.
Corollary 1Under the conditions of Theorem 1, we havefor.
Corollary 2If the assumptionoris replaced byor, i.e. , then the conclusion of Theorem 1 also holds.
Corollary 3If the assumptionis replaced byor, then problem (1.1)(1.3) admits a global weak solutionsuch that, andfor.
4 Invariant property and vacuum isolating of global solutions
In this section, we discuss the invariance of some sets under the flow of (1.1)(1.3) and vacuum isolating behavior of solutions for problem (1.1)(1.3).
4.1 Invariant property of global solutions
Theorem 2Assume thatpandqsatisfy (H), . If, are the two roots of the equation, then the following hold.
(1) All solutions of problem (1.1)(1.3) with initial energybelong tofor, provided thator.
(2) All solutions of problem (1.1)(1.3) with initial energybelong tofor, provided that.
Proof Firstly, we consider the case of . Here we denote .
(1) Let be any solution of problem (1.1)(1.3) with initial energy and or , T be the existence time of . Since
Next we prove for , . Otherwise, there exists a such that for some , i.e., or . From Lemma 7
we see that is impossible. On the other hand, if and , then by Lemma 4, we have , which contradicts (4.1).
(2) Let be any solution of problem (1.1)(1.3) with initial energy and , T be the existence time of . Since the sign of is not changed for , we have for . From this and for we obtain for .
Next we prove for and . Otherwise, there exists a such that for some , i.e. or . From (4.1) we see that is impossible. On the other hand, let be the first time such that , then for . From (4.1) and Lemma 2, we have for . Hence we have , thus by Lemma 4, we get , which contradicts (4.1).
For the case of , we can obtain the same results as the case by Lemma 4, we omit it here. □
Remark 2 Assume that , then and are invariant under the flow of (1.1)(1.3) for any .
From the above Theorem 2 and Lemma 2, we can easily get the following conclusions.
Theorem 3Letpandqsatisfy (H), . Assume that, are the two roots of the equation, then the following hold.
(1) All solutions of problem (1.1)(1.3) with initial energyandbelong tofor.
(2) All solutions of problem (1.1)(1.3) with initial energyandbelong tofor.
Remark 3 Let p and q satisfy (H), . Assume that , then and are invariant under the flow of (1.1)(1.3) for any .
4.2 Vacuum isolating of global solutions
The result of Theorem 3 shows that for any given , there exists a corresponding vacuum region of solutions
for the set of all solutions of problem (1.1)(1.3) with initial energy satisfying , there is no solution in and all solutions are isolated by . This phenomenon is called the phenomenon of vacuum isolating of solutions. Obviously, the vacuum region of solutions becomes bigger and bigger with decreasing of e. As the limit case , we obtain the biggest vacuum region of solutions (for )
Theorem 4Letpandqsatisfy (H), . All nontrivial solutions of problem (1.1)(1.3) with initial energylie outside of the ball (maybe in).
Theorem 5Letpandqsatisfy (H), . All nontrivial solutions of problem (1.1)(1.3) with initial energysatisfy
and
Remark 4 The proofs of Theorems 45 are similar to Theorems 4.74.8 in [20], we omit them.
5 Nonexistence of global solutions
In this section, we given the sufficient condition of global nonexistence of solutions.
Theorem 6Assume that, , is a local solution of problem (1.1)(1.3) on, then no solution of (1.1)(1.3) can exist onwhen.
Proof Assume for contradiction that there is a solution of (1.1)(1.3) on .
Define
hence
Multiplying (1.1) by u and integrating over Ω, we have
As , we see from (5.1) and (5.2) that
Since , we have . By the embedding theorem and (5.3), we have
This is impossible, since the left hand side is finite and the right hand side goes to ∞ as . □
6 Existence of global solution with critical initial conditions
In this section, we prove the result of global existence with critical initial conditions.
Theorem 7Assume thatpandqsatisfy (H), . If, or, , are the two roots of the equation, then problem (1.1)(1.3) admits a global solutionsuch that, andfor anyand, where.
Proof Let , , . Consider the initial condition
with the corresponding problem (1.1)(1.3) and suppose that are two roots of the equation .
Since , we obtain , i.e.. Thus . From and Lemma 1, we get
Obviously, implies that . As and is increasing with δ, it follows that .
Thus, by Theorem 1, the problem admits a global solution such that , and for any , satisfying
i.e.
Hence there exist u, ξ, and subsequence of such that
By using the monotone operator method, we get .
On the other hand, letting in we get in . Also , as .
Furthermore, from (6.2) and Lemma 7, we have for . On the other hand, from (6.3) we have
By Lemma 2, we have and for any . □
Remark 5 The invariant sets and vacuum of solutions for problem (1.1)(1.3) with critical initial conditions also occur.
Remark 6 Taking or , Theorem 7 is still satisfied and generalizes the results of [11]. Similarly, the invariant sets and vacuum isolating of solutions also occur.
Remark 7 In fact, all the results in our paper also hold for the homogeneous Dirichlet initial boundary value problem for the more general equation
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the manuscript and read and approved the final manuscript.
Acknowledgements
The second and third authors were supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2012005767) and the National Science Foundation of Shandong Province of China (ZR2012AM018) and the Fundamental Research Funds for the Central Universities (No. 201362032), respectively. The authors would like to express their sincere gratitude to the anonymous reviewers for their insightful and constructive comments.
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