Global existence and nonexistence of solutions for quasilinear parabolic equation
© Xu et al.; licensee Springer. 2014
Received: 30 October 2013
Accepted: 22 January 2014
Published: 7 February 2014
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.
Keywordsfamily of potential wells global existence nonexistence vacuum isolating critical value
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 non-Newton 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 non-Newtonian 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.
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  and their results were similar. Until Liu  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 [20–25]. For instance, Liu and Zhao  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 [10–12, 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.
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.
for , ;
where T is either infinity or the limit of the existence interval of solution.
where , .
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 (, Lemma 2.2)
There exists a unique such that ;
, i.e. is increasing for ; , i.e. is decreasing for ;
Lemma 2 (, Lemmas 2.1-2.3)
Let , then if and only if .
Let , then if and only if .
Let , then if and only if .
Lemma 3 (, Lemma 2.4)
takes the maximum at , where , ;
is increasing on and decreasing on ;
For any given , the equation has exactly two roots and .
Lemma 4 (, Lemma 2.5)
, where , , .
Proposition 1 , where , , .
Proof The result can easily be obtained by Lemma 4 and the fact that is equivalent to . □
Obviously, we have .
Remark 1 From , we see that implies that .
Obviously, we have .
Note that , hence for any given , when , we have and . This implies that , where satisfies .
Lemma 5 (, Theorem 2.7)
Lemma 6 (, Lemma 2.10)
Assume that for some given , are the two roots of the equation , then the sign of is not changed for .
Lemma 7 (, Lemma 2.8)
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.
If , then the solution is uniquely determined by the initial function;
If a.e. in Ω, the solution a.e. in Ω for any fixed , hence is a solution of the problem (1.1)-(1.3).
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.
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  we can easily get the results (1)-(3), here we omit the proofs. □
Similarly, we can get the following conclusions directly.
Corollary 1 Under the conditions of Theorem 1, we have for .
Corollary 2 If the assumption or is replaced by or , i.e. , then the conclusion of Theorem 1 also holds.
Corollary 3 If the assumption is replaced by or , then problem (1.1)-(1.3) admits a global weak solution such that , and for .
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
All solutions of problem (1.1)-(1.3) with initial energy belong to for , provided that or .
All solutions of problem (1.1)-(1.3) with initial energy belong to for , provided that .
Proof Firstly, we consider the case of . Here we denote .
by Theorem 1, we have .
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.
All solutions of problem (1.1)-(1.3) with initial energy and belong to for .
All solutions of problem (1.1)-(1.3) with initial energy and belong to for .
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
Theorem 4 Let p and q satisfy (H), . All nontrivial solutions of problem (1.1)-(1.3) with initial energy lie outside of the ball (maybe in ).
Remark 4 The proofs of Theorems 4-5 are similar to Theorems 4.7-4.8 in , we omit them.
5 Nonexistence of global solutions
In this section, we given the sufficient condition of global nonexistence of solutions.
Theorem 6 Assume that , , is a local solution of problem (1.1)-(1.3) on , then no solution of (1.1)-(1.3) can exist on when .
Proof Assume for contradiction that there is a solution of (1.1)-(1.3) on .
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 7 Assume that p and q satisfy (H), . If , or , , are the two roots of the equation , then problem (1.1)-(1.3) admits a global solution such that , and for any and , where .
with the corresponding problem (1.1)-(1.3) and suppose that are two roots of the equation .
Obviously, implies that . As and is increasing with δ, it follows that .
By using the monotone operator method, we get .
On the other hand, letting in we get in . Also , as .
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 . Similarly, the invariant sets and vacuum isolating of solutions also occur.
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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