A Remark on the Blowup of Solutions to the Laplace Equations with Nonlinear Dynamical Boundary Conditions
© Hongwei Zhang and Qingying Hu. 2010
Received: 24 April 2010
Accepted: 7 August 2010
Published: 11 August 2010
We present some sufficient conditions of blowup of the solutions to Laplace equations with semilinear dynamical boundary conditions of hyperbolic type.
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.
Hintermann  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. , Araruna et al. , and Hu et al. . In addition, Doronin et al.  studied a class hyperbolic problem with second-order 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 , when , by use of Jensen's inequality and Glassey's method . Kirane et al.  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].
Amann and Fila , Kirane , and Koleva and Vulkov  Vulkov  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  and the norm of is denoted by .
2. Blowup of the Solutions
and the boundary conditions are satisfied in the trace sense .
Lemma 2.1 (see ).
Suppose that and . Then, .
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.
since , where
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.
where are defined as in Section 1. Then, the solution of problem (1.1)–(1.4) blows up in a finite time.
and thus is monotonically increasing on This contradicts . Therefore, and hence as .
and as . The theorem is proved.
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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