Liouville theorems for a singular parabolic differential inequality with a gradient term
© Fang and Xu; licensee Springer. 2014
Received: 31 October 2013
Accepted: 30 January 2014
Published: 11 February 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.
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.
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.
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.
- (ii)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.
where and for sufficiently small .
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 some constant .
for some constant .
where is a parameter to be chosen later according to the nature of problem (1.1)-(1.2).
Then, by appropriate combinations of the above functions, one can choose suitable test functions and obtain the following nonlinear Liouville theorems.
- (I)When both and are singular on ∂ Ω, if
- (II)When both and are singular at 0, we consider the following two conditions: (i)(ii)
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 .
when both and are singular at 0.
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.
then any nonnegative weak solution to problem (1.1a)-(1.2) is trivial, i.e., a.e. in S.
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.
and hence we get a.e. in S.
Step 3. When both and are singular at , take .
Therefore, we obtain a.e. in S, and the proof is completed.
4 The proof of Theorem 2
- (i)Assume that both and are singular on ∂ Ω. It follows from (2.2), (3.7), and (3.10) that
- (ii)Assume that both and are singular at 0. By combining (2.3) and (3.7) with (3.14), we get the inequality
If or , the inequalities in (4.2) do not hold, which is a contradiction. The proof is completed.
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.
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