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

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

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 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 [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 .

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 right-hand 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

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.

1. Garipov, RM: On the linear theory of gravity waves: the theorem of existence and uniqueness. Archive for Rational Mechanics and Analysis. 24, 352–362 (1967 )

2. Hintermann, T: Evolution equations with dynamic boundary conditions. Proceedings of the Royal Society of Edinburgh. Section A. 113(1-2), 43–60 (1989). Publisher Full Text

3. Kirane, M: Blow-up for some equations with semilinear dynamical boundary conditions of parabolic and hyperbolic type. Hokkaido Mathematical Journal. 21(2), 221–229 (1992)

4. Lamb, H: Hydrodynamics, Cambridge University Press, Cambridge, Mass, USA (1916)

5. Langer, RE: A problem in diffusion or in the flow of heat for a solid in contact with a fluid. Tohoku Mathematical Journal. 35, 260–275 (1932)

6. Lions, J-L: Quelques méthodes de résolution des problèmes aux limites non linéaires,p. xx+554. Dunod, Paris, France (1969).

7. Cavalcanti, MM, Cavalcanti, VND: On solvability of solutions of degenerate nonlinear equations on manifolds. Differential and Integral Equations. 13(10–12), 1445–1458 (2000)

8. Cavalcanti, MM, Cavalcanti, VND: Existence and asymptotic stability for evolution problems on manifolds with damping and source terms. Journal of Mathematical Analysis and Applications. 291(1), 109–127 (2004). Publisher Full Text

9. Cavalcanti, MM, Cavalcanti, VND, Fukuoka, R, Soriano, JA: Asymptotic stability of the wave equation on compact surfaces and locally distributed damping—a sharp result. Transactions of the American Mathematical Society. 361(9), 4561–4580 (2009). Publisher Full Text

10. Cavalcanti, MM, Cavalcanti, VND, Fukuoka, R, Soriano, JA: Uniform stabilization of the wave equation on compact manifolds and locally distributed damping—a sharp result. Journal of Mathematical Analysis and Applications. 351(2), 661–674 (2009). Publisher Full Text

11. Cavalcanti, MM, Khemmoudj, A, Medjden, M: Uniform stabilization of the damped Cauchy-Ventcel problem with variable coefficients and dynamic boundary conditions. Journal of Mathematical Analysis and Applications. 328(2), 900–930 (2007). Publisher Full Text

12. Andrade, D, Cavalcanti, MM, Cavalcanti, VND, Oquendo, HP: Existence and asymptotic stability for viscoelastic evolution problems on compact manifolds. Journal of Computational Analysis and Applications. 8(2), 173–193 (2006)

13. Andrade, D, Cavalcanti, MM, Cavalcanti, VND, Oquendo, HP: Existence and asymptotic stability for viscoelastic evolution problems on compact manifolds. II. Journal of Computational Analysis and Applications. 8(3), 287–301 (2006)

14. Antunes, GO, Crippa, HR, da Silva, MDG: Periodic problem for a nonlinear-damped wave equation on the boundary. Mathematical Methods in the Applied Sciences. 33(11), 1275–1283 (2010)

15. Araruna, FD, Antunes, GO, Medeiros, LA: Semilinear wave equation on manifolds. Annales de la Faculté des Sciences de Toulouse. 11(1), 7–18 (2002). Publisher Full Text

16. Hu, Q-Y, Zhu, B, Zhang, H-W: A decay result to an elliptic equation with dynamical boundary condition. Chinese Quarterly Journal of Mathematics. 24(3), 365–369 (2009)

17. Doronin, GG, Larkin, NA, Souza, AJ: A hyperbolic problem with nonlinear second-order boundary damping. Electronic Journal of Differential Equations.(28), 1–10 (1998)

18. Glassey, RT: Blow-up theorems for nonlinear wave equations. Mathematische Zeitschrift. 132, 183–203 (1973). Publisher Full Text

19. Kirane, M, Nabana, E, Pohozaev, SI: Nonexistence of global solutions to an elliptic equation with a dynamical boundary condition. Boletim da Sociedade Paranaense de Matemática. 3rd Série. 22(2), 9–16 (2004)

20. Amann, H, Fila, M: A Fujita-type theorem for the Laplace equation with a dynamical boundary condition. Acta Mathematica Universitatis Comenianae. 66(2), 321–328 (1997)

21. Koleva, M, Vulkov, L: Blow-up of continuous and semidiscrete solutions to elliptic equations with semilinear dynamical boundary conditions of parabolic type. Journal of Computational and Applied Mathematics. 202(2), 414–434 (2007). Publisher Full Text

22. Vulkov, LG: Blow up for some quasilinear equations with dynamical boundary conditions of parabolic type. Applied Mathematics and Computation. 191(1), 89–99 (2007). Publisher Full Text

23. Belinsky, B: Eigenvalue problems for elliptic type partial differential operators with spectral parameters contained linearly in boundary conditions. In: Proceedings of the 8th International Symposium on Algorithms and Computation (ISAAC '97), December 1997, Singapore

24. Escher, J: Nonlinear elliptic systems with dynamic boundary conditions. Mathematische Zeitschrift. 210(3), 413–439 (1992)

25. Fila, M, Quittner, P: Global solutions of the Laplace equation with a nonlinear dynamical boundary condition. Mathematical Methods in the Applied Sciences. 20(15), 1325–1333 (1997). Publisher Full Text

26. Fila, M, Quittner, P: Large time behavior of solutions of a semilinear parabolic equation with a nonlinear dynamical boundary condition. Topics in Nonlinear Analysis, Progr. Nonlinear Differential Equations Appl., pp. 251–272. Birkhäuser, Basel, Switzerland (1999)

27. Koleva, M: On the computation of blow-up solutions of elliptic equations with semilinear dynamical boundary conditions. In: Proceedings of the 4th International Conference on Large-Scale Scientific Computing (LSSC '03), June 2003, Sozopol, Bulgaria, Lecture Notes in Computer Sciences. 2907, 105–123

28. Koleva, MN, Vulkov, LG: On the blow-up of finite difference solutions to the heat-diffusion equation with semilinear dynamical boundary conditions. Applied Mathematics and Computation. 161(1), 69–91 (2005). Publisher Full Text

29. Marinho, AO, Lourêdo, AT, Lima, OA: On a parabolic strongly nonlinear problem on manifolds. Electronic Journal of Qualitative Theory of Differential Equations.(13), 1–20 (2008)

30. Vitillaro, E: On the Laplace equation with non-linear dynamical boundary conditions. Proceedings of the London Mathematical Society. 93(2), 418–446 (2006). Publisher Full Text

31. Yin, Z: Global existence for elliptic equations with dynamic boundary conditions. Archiv der Mathematik. 81(5), 567–574 (2003). Publisher Full Text

32. Lions, JL, Magenes, E: Nonhomegeneous Boundary Value Problems and Applications, Springer, New York, NY, USA (1972)

33. Li, Y: Basic inequalityies and the uniqueness of the solutions for differential equations. Acta Scientiarum Naturalium Universitatis Jilinensis. 1, 257–293 (1960)