On Behavior of Solution of Degenerated Hyperbolic Equation

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.


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 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 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 .

Some Definitions and Auxiliary Results
Let us consider the equation 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: with the initial conditions 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 Ω and for any r > 0 and some fixed θ > 1 where η x, t ∈ W 1,1 p,ω Π a,b , η| Γ a,b 0, 0 < a < b.We will study the conditions at some generalized solution from Π 0 lim 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

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 ω x u 0 x dx 1. Denote g t Ω ω x u 0 x u x, t dx.Let's assume the fulfillment of the conditions: 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 > 0 in Ω 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

3.4
Let's formulate some transformations.After some simple manipulation we get

3.5
Using condition 3.1 , tending ε to zero at all t > 0, we obtain Here denoting we have Let's estimate first integral on the right-hand side of 3.8 .Using the Cauchy inequality with ε > 0 we get Hence So, from 3.8 we have

3.12
By virtue of Holder inequality we obtain

ISRN Applied Mathematics
Let's multiply 3.14 by g , then

3.16
Let's integrate by t from 0 to t, then from 3.16 we have 3.17

3.19
If the condition 3.2 is fulfilled, that is, Hence we have lim
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 .

3.22
Let's consider the function V x, t εe −λ 1 t/2 ϑ x .We have if ε > 0 is sufficiently 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 −λ 1 t/2 .Let us determine the class of functions

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 where T const > 0.
Proof.Analogously, as it is constructed in inequality 3.14 , we will obtain g t ≥ −λ 1 g C g e λ 1 σ g σ t .
Let one consider the following equation: , and let u 0 x > 0, Ω u 0 x dx 1, then one will have eigenfunction of the problem Δu λu 0 in Ω, u 0 on ∂Ω, and λ 1 > 0 the corresponding eigenvalue.
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

3.32
Using Δu 0 x ω x −λ 1 u 0 x ω x we will get 3.33 Using the results of paper 12 we will obtain the required result.

Theorem 3 . 3 .
Let one assume that |f x, t, u | ≤ c 3 c 4 t m |u| σ , σ > 1.There exists δ > 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 Ω ⊂ B R , where B R {x : |x| ≤ R}.Let ϑ > 0 in B R be an eigenfunction, corresponding to positive eigenvalue λ 1 of the boundary-value problem Let's consider first the integral on the right-hand side in 3.30 .Let ϕ x > 0, ψ x > 0 in Ω and sup pϕ x ⊂ Ω, sup ψ x ⊂ Ω. Denote the surface of degeneration of 3.27 , that is, the boundary of the solution sup u t, x ≡ θ t .Since u t, x ≡ 0 on ∂θ t and Ω/θ t from the Green's formula, we get ∂n t is a derivative on direction of external norms to ∂θ t .Since u σ 1 0 is on ∂θ t and by virtue of continuity of flow ∂u σ 1 /∂u t ≡ ∇u σ 1 • n t 0 at x ∈ ∂θ t .Therefore, the last two integrals in 3.31 are equal to zero.Then we will obtain Δωu 0 x dx.
Ω ω x u 0 x u x, t dx.