Research Article On Behavior of Solution of Degenerated Hyperbolic Equation

International Scholarly Research Network ISRN Applied Mathematics Volume 2012, Article ID 124936, 10 pages doi:10.5402/2012/124936 Research Article On Behavior of Solution of Degenerated Hyperbolic Equation Tahir Gadjiev, 1 Rafig Rasulov, 2 and Orkhan Aliev 1 1 Institute of Mathematics and Mechanics, Academy of Sciences, Baku, Azerbaijan 2 The Sheki Institute of Teachers, Sheki Az5, Azerbaijan Correspondence should be addressed to Rafig Rasulov, rafig rasulov@yahoo.com Received 10 August 2012; Accepted 13 September 2012 Academic Editors: A. Bellouquid and F. Ding Copyright q 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. 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 α-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 T>0 such that u x, t L R n, t T<. 1.1 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

2 ISRN Applied Mathematics 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 2 u n t u ω x 2 x j x j i,j 1 p 2 u f x, t, u, u i 2.1 in bounded domain R n,n 2with nonsmooth boundary. Let s denote Π a,b { x, t,x,a < t < b}, Γ a,b { x, t, x,a < t < b}. Π a Π a,, Γ a Γ a,. The functions f x, t, u, f x, t, u / u are continuous with respect to u uniformly in Π 0 {u : u M} at any M <, f x, t, 0 0, f/ u u 0 0. Besides, the function f is measurable by all arguments and does not decrease with respect to u. Let s assume the fulfillment of the following Dirichlet boundary condition: u 0, x, 2.2 with the initial conditions u t 0 ϕ x, u t t 0 ψ x, 2.3 in some domain Π 0,a, where ϕ x and ψ x are smooth functions. Let s assume that ω x is measurable, nonnegative function satisfying the conditions ω L 1,loc and for any r>0 and some fixed θ>1 B r ω 1/ θ 1 dx <, ess sup x B r ω x C 1 r n θ 1 B r ω 1/ θ 1 dx ) 1 θ, 2.4 where B r {x ; x <r}. From condition 2.4 it follows that ess sup c 1 r n ωdx x r B r 2.5 and that ω A θ,thatis, [ ] θ 1 ωdx ω 1/ θ 1 dx cr nθ. 2.6 B r B r

ISRN Applied Mathematics 3 Condition 2.6 is called Makhenkhoupt condition see 8. Besides, as in 8 we will assume that ω D μ,μ<1 p/n,thatis, ) ω B s s nμ ω B h c 1. 2.7 h Let s introduce the weight Sobolev space u W 1 p,ω ω x u p u p) 1/p dx), 2.8 for any s h>0, where ω B S B S ω x dx with the finite norm W 1 p,ω. As the generalized solution of problem 2.1 2.3 in Π a,b we will define the function u x, t W 1,1 p,ω Π a,b such that [ u t η t dx dt ut x, b η x, b u t x, a η x, a ] dx Π a,b n ω x u p 2 u η x dx dt f x, t, u η x, t dx dt, i x i x j Π a,b Π a,b i,j 1 2.9 where η x, t Wp,ω Π 1,1 a,b,η Γa,b 0, 0 <a<b. We will study the conditions at some generalized solution from Π 0 lim max u x, t, 2.10 t T 0 at some T const. At f x, t, u u u p 1, p > 1 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 ϕ x and ψ x are adequately big, then it holds 2.10. For small ϕ x and ψ x if lim t u x, t 0, then u <ce at,a const > 0, it doesn t depend on u. Let s construct the sufficient condition on u, at which any solution of problem 2.1 2.3 at f x, t, u, ϕ x 0,ψ x 0,ϕ x / 0,ψ x / 0 has blowup without limitation of smallness on ϕ x and ψ x. Let s formulate some auxiliary results from 9, 10, and let s determine the harmonic operator L p ) L p u div u p 2 u, p > 1. 2.11 Lemma 2.1 see 9. There exists the positive eigenvalue of spectral problem for the operator L p to which corresponds the positive in eigenfunction.

4 ISRN Applied Mathematics Lemma 2.2 see 10. Let u, ϑ Wp, 1 u ϑ on and for any nonnegative η W o 1 p L p u η xi dx L p ϑ η xi dx, 2.12 with η 0. Then on whole domain, u ϑ. 3. Main Results Let u 0 x > 0 be an eigenfunction of spectral problem for the operator L p corresponding to λ λ 1 > 0, ω x u 0 x dx 1. Denote g t ω x u 0 x u x, t dx. Let s assume the fulfillment of the conditions: I u ω x x i x u 0 i x i p 2 u p 2 u 0 x i ) u0 ω x i dx 0 3.1 g 0 ) 2 λ1 g 2 0 2c 1 σ 1 gσ 1 0 > 0. 3.2 Theorem 3.1. Let f x, t, u α 0 u σ 1 u at x, t Π 0,u 0, σ>1, α 0 const > 0. There exists k const such that if, u x, 0 0, u t x, 0 0, u x, 0 u 0dx k, u t x, 0 u 0dx k or condition 3.2 is fulfilled. Then lim t T max ω x u 0 x u x, t, 3.3 where T const > 0. Proof. Let s assume the opposite. Then u x, t is a solution of 2.1 in Π 0 and condition 2.2 is fulfilled on Γ 0. By the virtue of Lemma 2.2 in Π 0,u x, t > 0. Substitute in 2.9 η ε 1 u 0 x ω x, b a ε, a > 0, ε > 0, where u 0 x > 0in is an eigenfunction of spectral problem for the operator L p, corresponding to eigenvalue λ 1 > 0. Such eigenvalue exists by virtue of Lemma 2.1. As a result, we will obtain ε 1 [ ] ω x u t x, a ε u 0 x dx ω x u t x, a u 0 x dx n ε Π 1 ω x u p 2 u η a,a ε x dxdt ε 1 u 0 ωf x, t, u dxdt. i x i x j Π a,a ε i,j 1 3.4 Let s formulate some transformations. After some simple manipulation we get ε Π 1 ω x u 0 a,a ε x i p 2 u 0 x i η x j dxdt. 3.5

ISRN Applied Mathematics 5 Using condition 3.1, tending ε to zero at all t>0, we obtain u 0 x ω x u t x, t dx λ 1 u 2 0 t ω2 x dx u 0 ω u σ dx. 3.6 Here denoting g t u 0 x ω x u x, t dx, 3.7 we have g t λ 1 u 2 0 ω2 x dx a 0 u 0 ω u σ dx. 3.8 Let s estimate first integral on the right-hand side of 3.8. Using the Cauchy inequality with ε>0weget ωuu 0 dx ε u 2 dx 1 2 2ε ω 2 u 2 0 dx. 3.9 Hence ωuu 0 dx ε u 2 dx < 1 2 2ε ω 2 u 2 0 dx, 3.10 then λ 1 ω 2 u 2 0 dx > λ 1 ωuu 0 λ 1ε 2 u 2 dx > λ 1 ωuu 0 dx. 3.11 So, from 3.8 we have g t λ 1 ωu 2 0 udx a 0 u 0 ωu σ dx. 3.12 By virtue of Holder inequality we obtain uu 0 ωdx) σ [ ) 1/σ u σ u 0 ωdx ) σ 1 /σ ] σ ωu 0 dx c 1 u σ u 0 ωdx. 3.13 Thus, we have g t λ 1 g t c 1 g σ t, c const > 0. 3.14

6 ISRN Applied Mathematics Let s multiply 3.14 by g, then 1 g ) ) 2 c1 g σ g λ 1 gg c 1 g σ g 1 1 2 2 λ g 2). 3.15 Hence g ) 2 ) c1 g σ g λ 1 g 2). 3.16 Let s integrate by t from0tot, then from 3.16 we have t g ) ) 2 t t dt c1 g σ g dt λ 1 g 2) dt, 0 0 0 g t ) 2 g 0 ) 2 2c [ ] [ ] 1 g σ 1 t g σ 1 0 λ 1 g 2 t g 2 0, σ 1 3.17 or g t ) 2 2c 1 σ 1 gσ 1 t λ 1 g 2 t 2c 1 σ 1 gσ 1 0 λ 1 g 2 0 g 2 0. 3.18 Denote A g 0 ) 2 λ1 g 2 0 2c 1 σ 1 gσ 1 0. 3.19 If the condition 3.2 is fulfilled, that is, ) 2 ) 2 ω x u 0 x ψ x dx λ 1 ω x u 0 x ϕ x dx 2c στ 1 1 ω x u 0 x ϕ x dx) > 0 σ 1 3.20 and if g 0 >c 2 λ 1 / 2c 1 / σ 1 2/ σ 1, then lim t T 0 g t. Hence we have lim t T 0 max ω x u 0 x u x, t. 3.21 Remark 3.2. This theorem generalizes the corresponding result 7. That is why, 2.1 is not a solution in Π 0, satisfying the boundary condition 2.2, if u x, 0 0, u t x, 0 0 is not smaller. Now we will show that at small initial functions the solution of problem 2.1 2.3 exists in small in Π 0.

ISRN Applied Mathematics 7 Theorem 3.3. Let one assume that f x, t, u c 3 c 4 t m u σ,σ >1. Thereexistsδ>0 such that if ψ x <δ, ϕ x <δ, then the solution of problem 2.1 2.3 exists in small in Π 0 and u x, t c 5 e αt,α const > 0 does not depend on u. Proof. Let B R, where B R {x : x R}. Letϑ > 0inB R be an eigenfunction, corresponding to positive eigenvalue λ 1 of the boundary-value problem L p u λ 1 u 0, x, u 0, x. 3.22 Let s consider the function V x, t εe λ 1t/2 ϑ x. We have ) 1 V tt L p V f x, t, V ε 4 λ2 1 c 2 e λ1t/2 ϑ x c 3 c 4 t m ε σ e λ1t/2 ϑ σ 0, x, t Π 0, V > 0, x, t Γ 0, 3.23 if ε>0issufficiently small. Inequality 3.23 is understood in weak sense see 11. From 3.23 and Lemma 2.2, it follows that u V c 5 e λ1t/2. Let us determine the class of functions ϕ x δ, ψ x δ ε min ϑ x consisting from g x, t continuous in Π, equal to zero at t T and such that g x, t Ke ht is a subset at Banach space of continuous functions in Π, with the norm g sup Π, ge ht. Let θ t C R 1), θ t 1, if t T, θ t 0, if t>t 1. 3.24 Let s determine the operator H on K putting Hg θ t z, g K, where z is a solution of linearizing problem. By virtue of obtained estimation above, H maps K to K. This follows from the obtained estimation and theorem on the solution of the hyperbolic problems in Π a,a at the small 11. From Lerey-Shaudeer theorem, it follows that the operator H has a fixed point z.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 lim max t T 0 ω x u 0 x u x, t, 3.25 where T const > 0. Theorem 3.4. Let f x, t, u c σ e λ 1σt u σ at x, t Π 0,u 0, σ const > 1, and λ 1 be positive eigenvalue of problem 3.16 in which corresponds to the positive in eigenfunction. If u x, 0 0,u t x, 0 0,u x, 0 / 0,whereu t x, 0 / 0, u x, t 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 g t λ 1 g C g e λ 1σ g σ t. 3.26

8 ISRN Applied Mathematics Let g t ψ t e λ1t.from 3.25 it follows that ψ c 8 ψ τ. Hence ψ t at t T 0. Thus it tends g t to k and t T 0. Consequently, max ω x u x, t also tends to infinity. Corollary 3.5. Let f x, t, u c 8 e λ 1σ u σ at x, t Π 0,u 0,σ >1. Then there is no positive in Π 0 solution of the problem 2.1 2.3. Let one consider the following equation: 2 u n t ω x u σ u ) f x, t, u, 2 x i x j i,j 1 3.27 in bounded domain R n,n 2, and let u 0 x > 0, u 0 x dx 1, then one will have eigenfunction of the problem Δu λu 0 in, u 0 on, and λ 1 > 0 the corresponding eigenvalue. Theorem 3.6. Let f x, t, u α 0 u σ 1 at α 0 const 0 at x, t Π 0,u 0,σ > 1. There exists k const > 0 such that if u x, 0 0,u t x, 0 0, u x, 0 u 0dx k, u t x, 0 u 0dx k or condition 3.2 is fulfilled, then lim t T max ω x u 0 x u x, t, 3.28 where T const > 0. Proof. Let s assume a contrary. Then u x, t is a solution of 3.27 in Π 0 and condition 2.2 is fulfilled on Γ 0. Substituting in corresponding integral identity Γ 0 η ε 1 u 0 x ω x,b a ε, ε > 0toget ε 1 [ ] ω x u t x, a ε u 0 x dx ω x u t x, a u 0 x dx n ε 1 ω x u σ u )u 0 x ω x dx ε 1 x i x j Π a,a ε i,j 1 Π a,a ε u 0 ωf x, t, u dx dt, 3.29 where u 0 x > 0. Tending ε to zero, we ll obtain 2 t g t u 2 0 x ω x ωu σ u ) dx u 0 ωu σ 1 dx, x i x j 3.30 where g t ω x u 0 x u x, t dx. Let s consider first the integral on the right-hand side in 3.30. Letϕ x > 0, ψ x > 0in and sup pϕ x, sup ψ x. Denote the surface of

ISRN Applied Mathematics 9 degeneration of 3.27, that is, the boundary of the solution sup 0on θ t and /θ t from the Green s formula, we get u t, x θ t. Since u t, x ωu 0 x i ) ω σ u 1 4 dx ωu 0 Δu σ 1 dx x j σ 1 θ t [ 1 Δu 0 0 ω x σ 1 ud x σ 1 θ t θ t u 0 ω uσ 1 ds n t θ t ] u 0 ω u σ 1 dx, n t 3.31 where / n t is a derivative on direction of external norms to θ t. Since u σ 1 0ison θ t and by virtue of continuity of flow u σ 1 / u t u σ 1 n t 0atx θ t. Therefore, the last two integrals in 3.31 are equal to zero. Then we will obtain ωu 0 x ωu σ u ) 1 dx u σ 1 Δωu 0 x dx. x i x j σ 1 3.32 Using Δu 0 x ω x λ 1 u 0 x ω x we will get 2 g t t 2 1 λ ) 1 u σ 1 u 0 x ω x dx, t > 0. 3.33 σ 1 Using the results of paper 12 we will obtain the required result. References 1 S. Kaplan, On the growth of solutions of quasi-linear parabolic equations, Communications on Pure and Applied Mathematics, vol. 16, pp. 305 330, 1963. 2 H. Fujita, On the blowing up of solutions of the Cauchy problem for u t Δu u 1 α, the Faculty of Science, vol. 13, pp. 109 124, 1966. 3 K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic differential equations, Proceedings of the Japan Academy, vol. 49, pp. 503 505, 1973. 4 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. 5 K. Deng and H. A. Levine, The role of critical exponents in blow-up theorems: the sequel, Mathematical Analysis and Applications, vol. 243, no. 1, pp. 85 126, 2000. 6 A. A. Samarskii, V. A. Galaktinonov, S. P. Kurdyumov, and A. P. Mikahailov, Blow-Up in Quasi Linear Parabolic Equations, Nauka, Moscow, Russia, 1987. 7 Y. Ebihara, S. Kawashima, and H. A. Levine, On solutions to u tt x α Δu f u, Funkcialaj Ekvacioj, vol. 38, no. 3, pp. 539 544, 1995. 8 S. Chanillo and R. L. Wheeden, Weighted Poincaré and Sobolev inequalities and estimates for weighted Peano maximal functions, American Mathematics, vol. 107, no. 5, pp. 1191 1226, 1985. 9 P. Tolksdorf, On quasilinear boundary value problems in domains with corners, Nonlinear Analysis, vol. 5, no. 7, pp. 721 735, 1981. 10 D. Gilbarg and N. Trudinger, Elliptic Differential Equation With Partial Derivatives of the Second Order, Nauka, Moscow, Russia, 1989. 11 O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Uralceva, Lineynie i Kvazilineynie Uravneniya Parabolicheskogo Tipa, Nauka, Moscow, Russia, 1967.

10 ISRN Applied Mathematics 12 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.

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