ISRN Applied MathematicsVolume 2012 (2012), Article ID 124936, 10 pagesdoi:10.5402/2012/124936

Research Article

On Behavior of Solution of Degenerated Hyperbolic Equation

Tahir Gadjiev,¹ Rafig Rasulov,² and Orkhan Aliev¹

¹Institute of Mathematics and Mechanics, Academy of Sciences, Baku, Azerbaijan²The Sheki Institute of Teachers, Sheki Az5, Azerbaijan

Received 10 August 2012; Accepted 13 September 2012

Academic Editors: A. Bellouquid and F. Ding

Copyright © 2012 Tahir Gadjiev et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The purpose of this paper is to learn some features of hyperbolic type of nonlinear equations. It is shown that the solution of the equation approaches to the endlessness in the inside of some initial conditions and time of the special marks. The local existence of the equation’s solution has been proved and the problem of unlimited increasing on the solution of nonlinear hyperbolic equations type during the finite time is investigated.

1. Introduction

In this paper the unbounded increasing solution of the nonlinear hyperbolic-type equation for the finite times is considered. These type equations describe the processes of electron and ionic heat conductivity in plasma, diffusion of neutrons, and -particles, and so forth. Investigation of unbounded solution or regime of peaking solutions occurs in theory of nonlinear equations where one of the essential ideas is the representation called eigenfunction of nonlinear dissipative surroundings. It is well known that even a simple nonlinearity, subject to critical of exponent, the solution of nonlinear hyperbolic type equation for the finite time may increase unboundedly, that is, there is a number such that In [1] the existence of unbounded solution for finite time with a simple nonlinearity has been proved. [2] has shown that any nonnegative solution subject to critical exponent is unbounded increasing for the finite time. Similar results were obtained in [3] and corresponding theorems are called Fujita-Hayakawa’s theorems. More detailed reviews can be found in [4–6].

The paper is organizing as follows. In Section 2 we present some definitions and auxiliary results. In Section 3, we give the main results for type nonlinear equation, where blow up solutions are obtained and the local solution exists. This results generalize the corresponding result [7].

2. Some Definitions and Auxiliary Results

Let us consider the equation in bounded domain with nonsmooth boundary. Let's denote , . . The functions are continuous with respect to uniformly in at any . Besides, the function is measurable by all arguments and does not decrease with respect to . Let's assume the fulfillment of the following Dirichlet boundary condition: with the initial conditions in some domain , where and are smooth functions. Let's assume that is measurable, nonnegative function satisfying the conditions and for any and some fixed where . From condition (2.4) it follows that and that , that is, Condition (2.6) is called Makhenkhoupt condition (see [8]). Besides, as in [8] we will assume that , that is,

Let's introduce the weight Sobolev space for any , where with the finite norm .

As the generalized solution of problem (2.1)–(2.3) in we will define the function such that where .

We will study the conditions at some generalized solution from at some .

At and at linear main part there are many works, such as [1, 2] on investigation properties of solutions having (2.10). We will show that if and are adequately big, then it holds (2.10). For small and if then , it doesn't depend on . Let's construct the sufficient condition on , at which any solution of problem (2.1)–(2.3) at , has “blowup” (without limitation of smallness on and ).

Let's formulate some auxiliary results from [9, 10], and let's determine the harmonic operator

Lemma 2.1 (see [9]). There exists the positive eigenvalue of spectral problem for the operator to which corresponds the positive in eigenfunction.

Lemma 2.2 (see [10]). Let on and for any nonnegative with . Then on whole domain .

3. Main Results

Let be an eigenfunction of spectral problem for the operator corresponding to . Denote . Let's assume the fulfillment of the conditions:

Theorem 3.1. Let at . There exists such that if, or condition (3.2) is fulfilled. Then where .

Proof. Let's assume the opposite. Then is a solution of (2.1) in and condition (2.2) is fulfilled on . By the virtue of Lemma 2.2 in . Substitute in (2.9) , where in is an eigenfunction of spectral problem for the operator , corresponding to eigenvalue . Such eigenvalue exists by virtue of Lemma 2.1. As a result, we will obtain Let’s formulate some transformations. After some simple manipulation we get Using condition (3.1), tending to zero at all , we obtain Here denoting we have Let's estimate first integral on the right-hand side of (3.8). Using the Cauchy inequality with we get Hence then So, from (3.8) we have By virtue of Holder inequality we obtain Thus, we have Let's multiply (3.14) by , then Hence Let's integrate by from 0 to , then from (3.16) we have or Denote If the condition (3.2) is fulfilled, that is, and if , then . Hence we have

Remark 3.2. This theorem generalizes the corresponding result [7].That is why, (2.1) is not a solution in , satisfying the boundary condition (2.2), if is not smaller.Now we will show that at small initial functions the solution of problem (2.1)–(2.3) exists in small in .

Theorem 3.3. Let one assume that . There exists such that if , then the solution of problem (2.1)–(2.3) exists in small in and does not depend on .

Proof. Let , where . Let in be an eigenfunction, corresponding to positive eigenvalue of the boundary-value problem Let's consider the function . We have if is sufficiently small. Inequality (3.23) is understood in weak sense (see [11]). From (3.23) and Lemma 2.2, it follows that . Let us determine the class of functions consisting from continuous in equal to zero at and such that is a subset at Banach space of continuous functions in with the norm .Let Let's determine the operator on putting , where is a solution of linearizing problem. By virtue of obtained estimation above, maps to . This follows from the obtained estimation and theorem on the solution of the hyperbolic problems in at the small [11].From Lerey-Shaudeer theorem, it follows that the operator has a fixed point . This shows the existence of solution in the small. The theorem is proved.

Note that the sufficient condition at which any nonnegative solution of problem (2.1)–(2.3) has “blow-up” is where .

Theorem 3.4. Let at , and be positive eigenvalue of problem (3.16) in which corresponds to the positive in eigenfunction. If , where , is a solution of problem (2.1)–(2.3) then it holds (3.25).

Proof. Analogously, as it is constructed in inequality (3.14), we will obtain Let . From (3.25) it follows that . Hence at . Thus it tends to and . Consequently, also tends to infinity.

Corollary 3.5. Let at . Then there is no positive in solution of the problem (2.1)–(2.3). Let one consider the following equation: in bounded domain , and let , then one will have eigenfunction of the problem in on , and the corresponding eigenvalue.

Theorem 3.6. Let at at . There exists such that if or condition (3.2) is fulfilled, then where .

Proof. Let's assume a contrary. Then is a solution of (3.27) in and condition (2.2) is fulfilled on . Substituting in corresponding integral identity    to get where . Tending to zero, we'll obtain where . Let’s consider first the integral on the right-hand side in (3.30). Let in and . Denote the surface of degeneration of (3.27), that is, the boundary of the solution . Since on and from the Green's formula, we get where is a derivative on direction of external norms to . Since is on and by virtue of continuity of flow at . Therefore, the last two integrals in (3.31) are equal to zero. Then we will obtain Using we will get Using the results of paper [12] we will obtain the required result.

References

S. Kaplan, “On the growth of solutions of quasi-linear parabolic equations,” Communications on Pure and Applied Mathematics, vol. 16, pp. 305–330, 1963.

H. Fujita, “On the blowing up of solutions of the Cauchy problem for ,” Journal of the Faculty of Science, vol. 13, pp. 109–124, 1966.

K. Hayakawa, “On nonexistence of global solutions of some semilinear parabolic differential equations,” Proceedings of the Japan Academy, vol. 49, pp. 503–505, 1973.

V. A. Galaktionov and H. A. Levine, “A general approach to critical Fujita exponents in nonlinear parabolic problems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 34, no. 7, pp. 1005–1027, 1998.

K. Deng and H. A. Levine, “The role of critical exponents in blow-up theorems: the sequel,” Journal of Mathematical Analysis and Applications, vol. 243, no. 1, pp. 85–126, 2000.

A. A. Samarskii, V. A. Galaktinonov, S. P. Kurdyumov, and A. P. Mikahailov, Blow-Up in Quasi Linear Parabolic Equations, Nauka, Moscow, Russia, 1987.

Y. Ebihara, S. Kawashima, and H. A. Levine, “On solutions to ,” Funkcialaj Ekvacioj, vol. 38, no. 3, pp. 539–544, 1995.

S. Chanillo and R. L. Wheeden, “Weighted Poincaré and Sobolev inequalities and estimates for weighted Peano maximal functions,” American Journal of Mathematics, vol. 107, no. 5, pp. 1191–1226, 1985.

P. Tolksdorf, “On quasilinear boundary value problems in domains with corners,” Nonlinear Analysis, vol. 5, no. 7, pp. 721–735, 1981.

D. Gilbarg and N. Trudinger, Elliptic Differential Equation With Partial Derivatives of the Second Order, Nauka, Moscow, Russia, 1989.

O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Uralceva, Lineynie i Kvazilineynie Uravneniya Parabolicheskogo Tipa, Nauka, Moscow, Russia, 1967.

T. S. Hajiev and R. A. Rasulov, “Explosive solution of the nonlinear equation of a parabolic type,” Applied Mathematics Letters, vol. 24, no. 10, pp. 1676–1679, 2011.