We present some sufficient conditions of blowup of the solutions to Laplace equations with semilinear dynamical boundary conditions of hyperbolic type.
1. Introduction
Let be a bounded domain of with a smooth boundary where and are closed and disjoint and possesses positive measure. We consider the following problem:
where are constants, is the Laplace operator with respect to the space variables, and is the outer unit normal derivative to boundary . are given initial functions. For convenience, we take in this paper.
The problem (1.1)–(1.4) can be used as models to describe the motion of a fluid in a container or to describe the displacement of a fluid in a medium without gravity; see [1–5] for more information. In recent years, the problem has attracted a great deal of people. Lions [6] used the theory of maximal monotone operators to solve the existence of solution of the following problem:
Hintermann [2] used the theory of semigroups in Banach spaces to give the existence and uniqueness of the solution for problem (1.5)–(1.7). Cavalcanti et al. [7–11] studied the existence and asymptotic behavior of solutions evolution problem on manifolds. In this direction, the existence and asymptotic behavior of the related of evolution problem on manifolds has been also considered by Andrade et al. [12, 13], Antunes et al. [14], Araruna et al. [15], and Hu et al. [16]. In addition, Doronin et al. [17] studied a class hyperbolic problem with secondorder boundary conditions.
We will consider the blowup of the solution for problem (1.1)–(1.4) with nonlinear boundary source term . Blowup of the solution for problem (1.1)–(1.4) was considered by Kirane [3], when , by use of Jensen's inequality and Glassey's method [18]. Kirane et al. [19] concerned blowup of the solution for the Laplace equations with a hyperbolic type dynamical boundary inequality by the test function methods. In this paper, we present some sufficient conditions of blowup of the solutions for the problem (1.1)–(1.4) when is a bounded domain and can be a nonempty set. We use a different approach from those ones used in the prior literature [3, 19].
Another related problem to (1.1)–(1.4) is the following problem:
Amann and Fila [20], Kirane [3], and Koleva and Vulkov [21] Vulkov [22] considered blowup of the solution of problem (1.8)–(1.10). For more results concerning the related problem (1.8)–(1.10), we refer the reader to [3, 6, 19–31] and their references. In these papers, existence, boundedness, asymptotic behavior, and nonexistence of global solutions for problem (1.8)–(1.10) were studied.
In this paper, the definition of the usually space can be found in [32] and the norm of is denoted by .
2. Blowup of the Solutions
In this paper, we always assume that the initial data , and and that the problem (1.1)–(1.4) possesses a unique local weak solution [2, 3, 6] that is, is in the class
and the boundary conditions are satisfied in the trace sense [2].
Lemma 2.1 (see [33]).
Suppose that and . Then, .
Theorem 2.2.
Suppose that is a weak solution of problem (1.1)–(1.4) and satisfies:
()
()
where . Then, the solution of problem (1.1)–(1.4) blows up in a finite time.
Proof.
Denote
then from (1.1)–(1.4), we have
Hence
Let Using condition (1) of Theorem 2.2, we have
Observing that
we know from (2.5)–(2.7) that
It follows from (2.8) that
where From (2.8) and (2.10), we have
Using the inversion of the Hölder inequality, we obtain
Substituting (2.12) and (2.13) into (2.11), we have
Noticing that
we have
We see from (2.9) and (2.10) that as . Therefore, there is a such that
Multiplying both sides of (2.16) by and using (2.9), we get
where
From (2.18) we have
where Integrating (2.20) over , we arrive at
Observe that when , the righthand side of (2.21) approaches to positive infinity since for sufficiently large ; hence, there is a such that the right side of (2.21) is larger than or equal to zero when . We thus have
Extracting the square root of both sides of (2.22) and noticing that , we obtain
since , where
Consider the following initial value problem of the Bernoulli equation:
Solving the problem (2.24), we obtain the solution
where Obviously, , and for
From (2.10), we see that
as Take sufficiently large such that It follows from (2.26) and the condition of Theorem 2.2 that
Therefore,
By virtue of the continuity of and the theorem of the intermediate values, there is a constant such that Hence, as It follows from Lemma 2.1 that Thus, as The theorem is proved.
Theorem 2.3.
Suppose that is a convex function, , where is a real number , and is a weak solution of problem (1.1)–(1.4)
where is the normalized eigenfunction (i.e., ) corresponding the smallest eigenvalue of the following Steklov spectral problem [23]:
where are defined as in Section 1. Then, the solution of problem (1.1)–(1.4) blows up in a finite time.
Proof.
Let
Then, . It follows from (1.1)–(1.4) that satisfies
Using Green's formula, we have
where we have used (2.31) and the fact that is the eigenfunction of the problem (1.1)–(1.4), and are denoted as the expressions in the first and the second parenthesis, respectively. From (2.32), we have
If , it is clear that otherwise, by (1.3) and (2.33),
Therefore, (2.36) implies that , that is,
Now, (2.35) takes the form
From Jensen's inequality and the condition , we have
Substituting the above inequality into (2.40), we get
Since , from the continuity of , it follows that there is a right neighborhood of the point , in which and hence If there exists a point such that but then is monotonically increasing on It follows from (2.42) that on
and thus is monotonically increasing on This contradicts . Therefore, and hence as .
Multiplying both sides of (2.42) by and integrating the product over , we get
Since and
then Extracting the square root of both sides of (2.44), we have
Equation (2.46) means that the interval of the existence of is finite this, that is,
and as . The theorem is proved.
Remark 2.4.
The results of the above theorem hold when one considers (1.1)–(1.4) with more general elliptic operator, like
and the corresponding boundary conditions
Acknowledgments
The authors are very grateful to the referee's suggestions and comments. The authors are supported by National Natural Science Foundation of China and Foundation of Henan University of Technology.
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