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Liouville theorems for a singular parabolic differential inequality with a gradient term
Journal of Inequalities and Applications volume 2014, Article number: 62 (2014)
In this paper, we study proofs of some new Liouville theorems for a strongly p-coercive quasi-linear parabolic type differential inequality with a gradient term and singular variable coefficients. The proofs are based on the test function method developed by Mitidieri and Pohozaev.
We consider a singular quasi-linear parabolic differential inequality with a gradient term
and the initial condition is given by
where , () is a bounded domain with sufficiently smooth boundary or , the initial function is nonnegative, and the coefficients and are positive and singular on the boundary ∂ Ω or at the origin 0.
Let be a Carathéodory function and define an operator L as
for any such that . The operator L is said to be strongly p-coercive (S-p-C), if there exist three constants and such that the inequality
holds for all . For example, the Laplace operator is S-2-C and the p-Laplace operator is S-p-C.
Inequality (1.1) appears in many fields, such as fluid mechanics, biological species, and population dynamics, see [1–3]. From the perspective of fluid mechanics, (1.1) describes the non-Newtonian filtration phenomenon and the flow of gas in a porous medium, in which is called the source term and is called the gradient absorption term.
There have been many results on the nonexistence of nonnegative nontrivial global solutions for nonlinear parabolic type equations or inequalities (and systems), see [4–15] and references therein. In 1966, Fujita  studied the following Cauchy problem of the semilinear heat equation:
and he obtained the critical exponent on the existence versus nonexistence of nonnegative nontrivial global solution. From then on, there have been a number of extensions of Fujita’s results in many directions, and many scholars pay close attention to his results. The nonexistence of a global solution for nonlinear partial differential equations, which arises from the Liouville type theorem of harmonic function, is a nonlinear Liouville type theorem. Thus one can prove some properties of solutions in a bounded domain and this is a kind of essential reflection of blow-up or singularity theory as well, see  and references therein. In the whole space, Kartsatos  studied a Liouville type theorem on the solution for a quasi-linear parabolic equation without singular variable coefficient and gradient term. For a more general evolution and quasi-linear parabolic type inequality, see [7, 8] and references therein. By using the test function method, Mitidieri and Pohozaev [9, 10] studied the Liouville type theorems for elliptic and parabolic equations without gradient source term. Afterwards, by using the same method, Wei  obtained the nonexistence of the nonnegative nontrivial global weak solution for a semilinear parabolic inequality with a weight function , but without a gradient term. Note that the gradient term in the right-hand side of (1.1) will bring substantial difficulty to the research. In the whole space, Mitidieri and Pohozaev  put forward an important result for a differential inequality with a gradient nonlinearity; they considered the following quasi-linear elliptic inequality:
and they proved that it has only constant solutions if , , , , and . Later, Karisti and Filippucci did further research on general forms of (1.4) (cf. [13, 14]). Galakhov  discussed a strongly p-coercive elliptic, parabolic, and hyperbolic type differential inequality with a gradient nonlinearity of constant coefficient in bounded and unbounded domains. Moreover, they obtained a Liouville theorem by making use of the test function method. Recently, Li and Li [16, 17] extended their conclusions to the case of an elliptic type inequality with singular variable coefficients and obtained the nonexistence of solutions to that inequality and to the Harnack inequality. For singular parabolic problems in conical domains, refer to [18–20] and the references therein.
Besides the works mentioned above, there are few studies on nonlinear Liouville theorems for a parabolic type differential inequality (1.1) with singular variable coefficients and a gradient term. The purpose of this paper is to investigate the influence of an S-p-C operator, the exponent of singular coefficients, and the gradient nonlinearity on the nonexistence of a nonnegative nontrivial global weak solution. The main difficulty lies in choosing a suitable test function according to different singularities of , , and inequality (1.1) with gradient nonlinearity. We will use the test function technique, developed by Mitidieri and Pohozaev [9, 10, 12], to show some Liouville theorems on the weak solution for problem (1.1)-(1.2) with bounded and unbounded domains. In particular, we recall that no use of comparison results and no conditions at infinity on the solution are required.
This paper is organized as follows: In Section 2, some preliminaries, such as some basic definitions and assumptions, and our main results are given. We prove the main results in Sections 3 and 4.
2 Preliminaries and main results
Since and inequality (1.1) is degenerate or singular, problem (1.1)-(1.2) has no classical solution in general, and so it is reasonable to consider other solutions. We state the two definitions of a weak solution and of a solution with a positive lower bound.
Definition 1 A nonnegative function is called a weak solution of problem (1.1)-(1.2) in , if the following two conditions are satisfied:
for any nonnegative function , we have(2.1)
Definition 2 A nonnegative function such that is said to be a solution with positive lower bound in S, if there exists such that a.e. in S.
We consider two types of singularities.
Type 1: Both and are singular on the boundary ∂ Ω, i.e., there exist constants and such that
where and for sufficiently small .
Type 2: Both and are singular at the origin in the case that , i.e., there exist constants and such that
To establish a priori estimates of the solutions, we define some test functions which have the form of a separation of the variables, and those functions will be widely used in the sequel.
For the space variable and singular Type 1, we introduce a cut-off function that satisfies
for some constant .
For the space variable and singular Type 2, we consider a cut-off function that satisfies
for some constant .
where is a parameter to be chosen later according to the nature of problem (1.1)-(1.2).
For the time variable, assume that and
Then, by appropriate combinations of the above functions, one can choose suitable test functions and obtain the following nonlinear Liouville theorems.
Theorem 1 Let , , and let . Suppose that A is S-p-C and (or ) are continuous functions that satisfy (2.2) (or (2.3)).
When both and are singular on ∂ Ω, if
then any nonnegative weak solution to problem (1.1)-(1.2) is trivial, i.e., a.e. in S.
When both and are singular at 0, we consider the following two conditions: (i)
If one of the conditions (i) and (ii) holds, then any nonnegative weak solution to problem (1.1)-(1.2) is trivial, i.e., a.e. in S, where θ is a parameter and .
Theorem 2 Suppose that the assumptions of Theorem 1 are satisfied. Then problem (1.1)-(1.2) has no solution with positive lower bound in S, if
when both and are singular on ∂ Ω, or if
when both and are singular at 0.
Remark 1 If inequality (1.1) is modified with
one can obtain the following results by a similar argument in the proof of Theorem 1.
Suppose that the assumptions of Theorem 1 are satisfied.
(I′) When both and are singular on ∂ Ω, if
then any nonnegative weak solution to problem (1.1a)-(1.2) is trivial, i.e., a.e. in S.
(II′) When both and are singular at 0, we consider the following two conditions: (i′)
If one of the conditions (i′) and (ii′) holds, then any nonnegative weak solution to problem (1.1a)-(1.2) is trivial, i.e., a.e. in S, where θ is a parameter and .
Remark 2 Since Theorem 2 is independent of the parameter θ, one can also obtain the same results for problem (1.1a)-(1.2).
3 The proof of Theorem 1
Our proof mainly consists of three steps.
Step 1. For , define and let be a cut-off function, where is defined as (2.6) and . Clearly, we get
For , which will be determined later, and , multiplying both sides of (1.1) with a test function and integrating the result over , we have
It follows from (1.3) that
By the definition of ψ, we get
By applying Young’s inequality to the three terms in the right-hand side of (3.1), we obtain
Since , , , and , we know that
Applying Young’s inequality to the two terms in the right-hand side of (3.3) and (3.4) yields
Let k be large enough such that . From inequalities (3.1)-(3.6), we deduce
We then have
It can be seen that , and we take . We then obtain
Step 2. When both and are singular on ∂ Ω, we take in (3.7). Evidently, one has
By (2.2), we get
By a standard change of variables and (2.4), we then obtain
By the conditions in (I) of Theorem 1, we have , , and . Then, letting and combining with a random selection of T yield
and hence we get a.e. in S.
Step 3. When both and are singular at , take .
Choosing in (3.7), one has
It follows from (2.3) that
Combining the above result with (2.5), we get
If one of the conditions (i) and (ii) in (II) of Theorem 1 holds, we have , , and . Then, letting and combining with a random selection of T, we finally arrive at
Therefore, we obtain a.e. in S, and the proof is completed.
4 The proof of Theorem 2
In this proof, we use the method of contradiction. We take the same test function as in Section 3 and the proof is similar to that of Theorem 1. Suppose there exists u such that .
Assume that both and are singular on ∂ Ω. It follows from (2.2), (3.7), and (3.10) that
By the definition of χ and the assumption , we have
i.e., one of following inequalities holds:
Since , if or , and is large enough and ε is sufficiently small, all inequalities in (4.1) do not hold, which is a contradiction.
Assume that both and are singular at 0. By combining (2.3) and (3.7) with (3.14), we get the inequality
Combining this inequality with , we have
i.e., . Clearly, we have the inequalities
If or , the inequalities in (4.2) do not hold, which is a contradiction. The proof is completed.
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This work is supported by the Natural Science Foundation of Shandong Province of China (ZR2012AM018) and the Fundamental Research Funds for the Central Universities (No. 201362032). The authors would like to sincerely thank all the reviewers for their insightful and constructive comments.
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript and read and approved the final manuscript.
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Fang, Z.B., Xu, L. Liouville theorems for a singular parabolic differential inequality with a gradient term. J Inequal Appl 2014, 62 (2014). https://doi.org/10.1186/1029-242X-2014-62
- Weak Solution
- Gradient Term
- Differential Inequality
- Liouville Theorem
- Liouville Type Theorem