Open Access

A Remark on the Blowup of Solutions to the Laplace Equations with Nonlinear Dynamical Boundary Conditions

Boundary Value Problems20102010:203248

DOI: 10.1155/2010/203248

Received: 24 April 2010

Accepted: 7 August 2010

Published: 11 August 2010

Abstract

We present some sufficient conditions of blowup of the solutions to Laplace equations with semilinear dynamical boundary conditions of hyperbolic type.

1. Introduction

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq1_HTML.gif be a bounded domain of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq2_HTML.gif with a smooth boundary https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq3_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq4_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq5_HTML.gif are closed and disjoint and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq6_HTML.gif possesses positive measure. We consider the following problem:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ1_HTML.gif
(1.1)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ2_HTML.gif
(1.2)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ3_HTML.gif
(1.3)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ4_HTML.gif
(1.4)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq7_HTML.gif are constants, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq8_HTML.gif is the Laplace operator with respect to the space variables, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq9_HTML.gif is the outer unit normal derivative to boundary https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq10_HTML.gif . https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq11_HTML.gif are given initial functions. For convenience, we take https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq12_HTML.gif 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 [15] 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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ5_HTML.gif
(1.5)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ6_HTML.gif
(1.6)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ7_HTML.gif
(1.7)

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. [711] 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 second-order boundary conditions.

We will consider the blowup of the solution for problem (1.1)–(1.4) with nonlinear boundary source term https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq13_HTML.gif . Blowup of the solution for problem (1.1)–(1.4) was considered by Kirane [3], when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq14_HTML.gif , 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq15_HTML.gif is a bounded domain and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq16_HTML.gif 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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ8_HTML.gif
(1.8)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ9_HTML.gif
(1.9)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ10_HTML.gif
(1.10)

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, 1931] 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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq17_HTML.gif can be found in [32] and the norm of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq18_HTML.gif is denoted by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq19_HTML.gif .

2. Blowup of the Solutions

In this paper, we always assume that the initial data https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq20_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq21_HTML.gif and that the problem (1.1)–(1.4) possesses a unique local weak solution [2, 3, 6] that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq22_HTML.gif is in the class
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ11_HTML.gif
(2.1)

and the boundary conditions are satisfied in the trace sense [2].

Lemma 2.1 (see [33]).

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq23_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq24_HTML.gif . Then, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq25_HTML.gif .

Theorem 2.2.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq26_HTML.gif is a weak solution of problem (1.1)–(1.4) and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq27_HTML.gif satisfies:

(1) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq29_HTML.gif

(2) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq31_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq32_HTML.gif

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq33_HTML.gif . Then, the solution of problem (1.1)–(1.4) blows up in a finite time.

Proof.

Denote
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ12_HTML.gif
(2.2)
then from (1.1)–(1.4), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ13_HTML.gif
(2.3)
Hence
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ14_HTML.gif
(2.4)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq34_HTML.gif Using condition (1) of Theorem 2.2, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ15_HTML.gif
(2.5)
Observing that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ16_HTML.gif
(2.6)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ17_HTML.gif
(2.7)
we know from (2.5)–(2.7) that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ18_HTML.gif
(2.8)
It follows from (2.8) that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ19_HTML.gif
(2.9)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ20_HTML.gif
(2.10)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq35_HTML.gif From (2.8) and (2.10), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ21_HTML.gif
(2.11)
Using the inversion of the Hölder inequality, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ22_HTML.gif
(2.12)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ23_HTML.gif
(2.13)
Substituting (2.12) and (2.13) into (2.11), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ24_HTML.gif
(2.14)
Noticing that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ25_HTML.gif
(2.15)
we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ26_HTML.gif
(2.16)
We see from (2.9) and (2.10) that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq36_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq37_HTML.gif . Therefore, there is a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq38_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ27_HTML.gif
(2.17)
Multiplying both sides of (2.16) by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq39_HTML.gif and using (2.9), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ28_HTML.gif
(2.18)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ29_HTML.gif
(2.19)
From (2.18) we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ30_HTML.gif
(2.20)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq40_HTML.gif Integrating (2.20) over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq41_HTML.gif , we arrive at
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ31_HTML.gif
(2.21)
Observe that when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq42_HTML.gif , the right-hand side of (2.21) approaches to positive infinity since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq43_HTML.gif for sufficiently large https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq44_HTML.gif ; hence, there is a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq45_HTML.gif such that the right side of (2.21) is larger than or equal to zero when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq46_HTML.gif . We thus have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ32_HTML.gif
(2.22)
Extracting the square root of both sides of (2.22) and noticing that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq47_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ33_HTML.gif
(2.23)

since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq48_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq49_HTML.gif

Consider the following initial value problem of the Bernoulli equation:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ34_HTML.gif
(2.24)
Solving the problem (2.24), we obtain the solution
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ35_HTML.gif
(2.25)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq50_HTML.gif Obviously, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq51_HTML.gif , and for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq52_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ36_HTML.gif
(2.26)
From (2.10), we see that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ37_HTML.gif
(2.27)
as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq53_HTML.gif Take https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq54_HTML.gif sufficiently large such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq55_HTML.gif It follows from (2.26) and the condition of Theorem 2.2 that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ38_HTML.gif
(2.28)
Therefore,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ39_HTML.gif
(2.29)

By virtue of the continuity of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq56_HTML.gif and the theorem of the intermediate values, there is a constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq57_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq58_HTML.gif Hence, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq59_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq60_HTML.gif It follows from Lemma 2.1 that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq61_HTML.gif Thus, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq62_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq63_HTML.gif The theorem is proved.

Theorem 2.3.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq64_HTML.gif is a convex function, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq65_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq66_HTML.gif is a real number https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq67_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq68_HTML.gif is a weak solution of problem (1.1)–(1.4)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ40_HTML.gif
(2.30)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq69_HTML.gif is the normalized eigenfunction (i.e., https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq70_HTML.gif ) corresponding the smallest eigenvalue https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq71_HTML.gif of the following Steklov spectral problem [23]:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ41_HTML.gif
(2.31)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ42_HTML.gif
(2.32)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ43_HTML.gif
(2.33)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq72_HTML.gif are defined as in Section 1. Then, the solution of problem (1.1)–(1.4) blows up in a finite time.

Proof.

Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ44_HTML.gif
(2.34)
Then, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq73_HTML.gif . It follows from (1.1)–(1.4) that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq74_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ45_HTML.gif
(2.35)
Using Green's formula, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ46_HTML.gif
(2.36)
where we have used (2.31) and the fact that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq75_HTML.gif is the eigenfunction of the problem (1.1)–(1.4), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq76_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq77_HTML.gif are denoted as the expressions in the first and the second parenthesis, respectively. From (2.32), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ47_HTML.gif
(2.37)
If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq78_HTML.gif , it is clear that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq79_HTML.gif otherwise, by (1.3) and (2.33),
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ48_HTML.gif
(2.38)
Therefore, (2.36) implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq80_HTML.gif , that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ49_HTML.gif
(2.39)
Now, (2.35) takes the form
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ50_HTML.gif
(2.40)
From Jensen's inequality and the condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq81_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ51_HTML.gif
(2.41)
Substituting the above inequality into (2.40), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ52_HTML.gif
(2.42)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq82_HTML.gif , from the continuity of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq83_HTML.gif , it follows that there is a right neighborhood https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq84_HTML.gif of the point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq85_HTML.gif , in which https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq86_HTML.gif and hence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq87_HTML.gif If there exists a point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq88_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq89_HTML.gif but https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq90_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq91_HTML.gif is monotonically increasing on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq92_HTML.gif It follows from (2.42) that on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq93_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ53_HTML.gif
(2.43)

and thus https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq94_HTML.gif is monotonically increasing on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq95_HTML.gif This contradicts https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq96_HTML.gif . Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq97_HTML.gif and hence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq98_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq99_HTML.gif .

Multiplying both sides of (2.42) by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq100_HTML.gif and integrating the product over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq101_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ54_HTML.gif
(2.44)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq102_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ55_HTML.gif
(2.45)
then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq103_HTML.gif Extracting the square root of both sides of (2.44), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ56_HTML.gif
(2.46)
Equation (2.46) means that the interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq104_HTML.gif of the existence of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq105_HTML.gif is finite this, that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ57_HTML.gif
(2.47)

and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq106_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_IEq107_HTML.gif . 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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ58_HTML.gif
(2.48)
and the corresponding boundary conditions
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F203248/MediaObjects/13661_2010_Article_902_Equ59_HTML.gif
(2.49)

Declarations

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.

Authors’ Affiliations

(1)
Department of Mathematics, Henan University of Technology

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© Hongwei Zhang and Qingying Hu. 2010

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