On Behavior of Solution of Degenerated Hyperbolic Equation

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 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∞(𝑅𝑛)}→∞,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 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 𝜕²𝑢𝜕𝑡²=^𝑛_𝑖,𝑗=1𝜕𝜕𝑥_𝑗𝜔(𝑥)𝜕𝑢𝜕𝑥_𝑗^𝑝−2𝜕𝑢𝜕𝑢_𝑖+𝑓(𝑥,𝑡,𝑢), (2.1) in bounded domain Ω⊂𝑅^𝑛,𝑛≥2 with nonsmooth boundary. Let's denote Π_𝑎,𝑏={(𝑥,𝑡),𝑥∈Ω,𝑎<𝑡<𝑏},Γ_𝑎,𝑏={(𝑥,𝑡), 𝑥∈𝜕Ω,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 Π₀×{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∈𝜕Ω, (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(Ω) and for any 𝑟>0 and some fixed 𝜃>1_𝐵𝑟𝜔^{−1/(𝜃−1)}𝑑𝑥<∞,esssup_{𝑥∈𝐵𝑟}𝜔(𝑥)≤𝐶₁𝑟^{𝑛(𝜃−1)}_𝐵𝑟𝜔^{−1/(𝜃−1)}𝑑𝑥^1−𝜃, (2.4) where 𝐵_𝑟={𝑥∈Ω;|𝑥|<𝑟}. From condition (2.4) it follows that esssup_{𝑥∈Ω𝑟}≤𝑐₁𝑟^−𝑛_𝐵𝑟𝜔𝑑𝑥 (2.5) and that 𝜔∈𝐴_𝜃, that is, _𝐵𝑟𝜔𝑑𝑥_𝐵𝑟𝜔^{−1/(𝜃−1)}𝑑𝑥^𝜃−1≤𝑐𝑟^𝑛𝜃. (2.6) Condition (2.6) is called Makhenkhoupt condition (see [8]). Besides, as in [8] we will assume that 𝜔∈𝐷_𝜇,𝜇<1+𝑝/𝑛, that is, 𝜔𝐵_𝑠𝜔𝐵_ℎ≤𝑐₁𝑠ℎ^𝑛𝜇. (2.7)

Let's introduce the weight Sobolev space ‖𝑢‖_𝑊¹_{𝑝,𝜔(Ω)}=_Ω𝜔(𝑥)|𝑢|^𝑝+|∇𝑢|^𝑝𝑑𝑥^1/𝑝, (2.8) for any 𝑠≥ℎ>0, where 𝜔(𝐵_𝑆)=∫_𝐵𝑆𝜔(𝑥)𝑑𝑥 with the finite norm 𝑊¹_𝑝,𝜔(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 _Π𝑎,𝑏𝑢_𝑡𝜂_𝑡𝑑𝑥𝑑𝑡+_Ω𝑢_𝑡(𝑥,𝑏)𝜂(𝑥,𝑏)−𝑢_𝑡(𝑥,𝑎)𝜂(𝑥,𝑎)𝑑𝑥+_Π𝑎,𝑏^𝑛_𝑖,𝑗=1𝜔(𝑥)𝜕𝑢𝜕𝑥_𝑖^𝑝−2𝜕𝑢𝜕𝑥_𝑖𝜕𝜂𝜕𝑥_𝑗𝑑𝑥𝑑𝑡=_Π𝑎,𝑏𝑓(𝑥,𝑡,𝑢)𝜂(𝑥,𝑡)𝑑𝑥𝑑𝑡, (2.9) where 𝜂(𝑥,𝑡)∈𝑊^1,1_𝑝,𝜔(Π_𝑎,𝑏),𝜂|_Γ𝑎,𝑏=0,0<𝑎<𝑏.

We will study the conditions at some generalized solution from Π₀lim_{𝑡→𝑇−0}max_Ω|𝑢(𝑥,𝑡)|=+∞, (2.10) 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_𝑝𝐿_𝑝𝑢=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 Ω eigenfunction.

Lemma 2.2 (see [10]). Let 𝑢,𝜗∈𝑊¹_𝑝(Ω),𝑢≤𝜗 on 𝜕Ω and for any nonnegative 𝜂∈𝑊^𝑜1_𝑝(Ω)_Ω𝐿_𝑝(𝑢)𝜂_𝑥𝑖𝑑𝑥≤_Ω𝐿_𝑝(𝜗)𝜂_𝑥𝑖𝑑𝑥, (2.12) with 𝜂≥0. Then on whole domain Ω,𝑢≤𝜗.

3. Main Results

Let 𝑢₀(𝑥)>0 be an eigenfunction of spectral problem for the operator 𝐿_𝑝 corresponding to 𝜆=𝜆₁>0,∫_Ω𝜔(𝑥)𝑢₀(𝑥)𝑑𝑥=1. Denote 𝑔(𝑡)=∫_Ω𝜔(𝑥)𝑢₀(𝑥)𝑢(𝑥,𝑡)𝑑𝑥. Let's assume the fulfillment of the conditions: 𝐼=_Ω𝜔(𝑥)𝜕𝑢𝜕𝑥_𝑖^𝑝−2𝜕𝑢𝜕𝑥_𝑖−𝜕𝑢₀𝜕𝑥_𝑖^𝑝−2𝜕𝑢₀𝜕𝑥_𝑖𝜕𝑢₀𝜔𝜕𝑥_𝑖𝑔𝑑𝑥≥0_′(0)²+𝜆₁𝑔²(0)−2𝑐₁𝑔𝜎+1^𝜎+1(0)>0. (3.1)(3.2)

Theorem 3.1. Let 𝑓(𝑥,𝑡,𝑢)≥𝛼₀|𝑢|^𝜎−1𝑢 at (𝑥,𝑡)∈Π₀,𝑢≥0,𝜎>1,𝛼₀=𝑐𝑜𝑛𝑠𝑡>0. There exists 𝑘=𝑐𝑜𝑛𝑠𝑡 such that if, 𝑢(𝑥,0)≥0,𝑢^′_𝑡(𝑥,0)≥0,∫_Ω𝑢(𝑥,0)𝑢₀𝑑𝑥≥𝑘,∫_Ω𝑢^′_𝑡(𝑥,0)𝑢₀𝑑𝑥≥𝑘 or condition (3.2) is fulfilled. Then lim_𝑡→𝑇max_Ω𝜔(𝑥)⋅𝑢₀(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 Π₀ and condition (2.2) is fulfilled on Γ₀. By the virtue of Lemma 2.2 in Π₀,𝑢(𝑥,𝑡)>0. Substitute in (2.9) 𝜂=𝜀⁻¹𝑢₀(𝑥)𝜔(𝑥),𝑏=𝑎+𝜀,𝑎>0,𝜀>0, where 𝑢₀(𝑥)>0 in Ω is an eigenfunction of spectral problem for the operator 𝐿_𝑝, corresponding to eigenvalue 𝜆₁>0. Such eigenvalue exists by virtue of Lemma 2.1. As a result, we will obtain 𝜀⁻¹_Ω𝜔(𝑥)𝑢_𝑡(𝑥,𝑎+𝜀)𝑢₀(𝑥)𝑑𝑥−_Ω𝜔(𝑥)𝑢_𝑡(𝑥,𝑎)𝑢₀(𝑥)𝑑𝑥+𝜀⁻¹_{Π𝑎,𝑎+𝜀}𝜔(𝑥)^𝑛_𝑖,𝑗=1𝜕𝑢𝜕𝑥_𝑖^𝑝−2𝜕𝑢𝜕𝑥_𝑖𝜕𝜂𝜕𝑥_𝑗𝑑𝑥𝑑𝑡=𝜀⁻¹_{Π𝑎,𝑎+𝜀}𝑢₀𝜔𝑓(𝑥,𝑡,𝑢)𝑑𝑥𝑑𝑡. (3.4)
Let’s formulate some transformations. After some simple manipulation we get 𝜀⁻¹_{Π𝑎,𝑎+𝜀}𝜔(𝑥)𝜕𝑢₀𝜕𝑥_𝑖^𝑝−2𝜕𝑢₀𝜕𝑥_𝑖𝜕𝜂𝜕𝑥_𝑗𝑑𝑥𝑑𝑡. (3.5) Using condition (3.1), tending 𝜀 to zero at all 𝑡>0, we obtain 𝜕𝜕𝑡_Ω𝑢₀(𝑥)𝜔(𝑥)𝑢_𝑡(𝑥,𝑡)𝑑𝑥≥−𝜆_1Ω𝑢²₀𝜔²(𝑥)𝑑𝑥+_Ω𝑢₀𝜔|𝑢|^𝜎𝑑𝑥. (3.6) Here denoting 𝑔(𝑡)=_Ω𝑢₀(𝑥)𝜔(𝑥)𝑢(𝑥,𝑡)𝑑𝑥, (3.7) we have 𝑔^′′(𝑡)≥−𝜆_1Ω𝑢²₀𝜔²(𝑥)𝑑𝑥+𝑎_0Ω𝑢₀𝜔|𝑢|^𝜎𝑑𝑥. (3.8) Let's estimate first integral on the right-hand side of (3.8). Using the Cauchy inequality with 𝜀>0 we get _Ω𝜔𝑢𝑢₀𝜀𝑑𝑥≤2_Ω𝑢²1𝑑𝑥+2𝜀_Ω𝜔²𝑢²₀𝑑𝑥. (3.9) Hence _Ω𝜔𝑢𝑢₀𝜀𝑑𝑥−2_Ω𝑢²1𝑑𝑥<2𝜀_Ω𝜔²𝑢²₀𝑑𝑥, (3.10) then 𝜆_1Ω𝜔²𝑢²₀𝑑𝑥>−𝜆_1Ω𝜔𝑢𝑢₀+𝜆₁𝜀2_Ω𝑢²𝑑𝑥>−𝜆_1Ω𝜔𝑢𝑢₀𝑑𝑥. (3.11) So, from (3.8) we have 𝑔^′′(𝑡)≥−𝜆_1Ω𝜔𝑢²₀𝑢𝑑𝑥+𝑎_0Ω𝑢₀𝜔𝑢^𝜎𝑑𝑥. (3.12) By virtue of Holder inequality we obtain _Ω𝑢𝑢₀𝜔𝑑𝑥^𝜎≤_Ω𝑢^𝜎𝑢₀𝜔𝑑𝑥^1/𝜎_Ω𝜔𝑢₀𝑑𝑥^{(𝜎−1)/𝜎𝜎}≤𝑐_1Ω𝑢^𝜎𝑢₀𝜔𝑑𝑥. (3.13) Thus, we have 𝑔^′′(𝑡)≥−𝜆₁𝑔(𝑡)+𝑐₁𝑔^𝜎(𝑡),𝑐=const>0. (3.14) Let's multiply (3.14) by 𝑔^′, then 12𝑔^′2′≥𝑐₁𝑔^𝜎𝑔^′−𝜆₁𝑔𝑔^′=𝑐₁𝑔^𝜎𝑔^′−12𝜆₁𝑔^2′. (3.15) Hence 𝑔^′2′≥𝑐₁𝑔^𝜎𝑔^′−𝜆₁𝑔^2′. (3.16) Let's integrate by 𝑡 from 0 to 𝑡, then from (3.16) we have ^𝑡₀𝑔^′2′𝑑𝑡≥𝑐₁^𝑡₀𝑔^𝜎𝑔^′𝑑𝑡−𝜆₁^𝑡₀𝑔^2′𝑔𝑑𝑡,_′(𝑡)²−𝑔^′(0)²≥2𝑐₁𝑔𝜎+1^𝜎+1(𝑡)−𝑔^𝜎+1(0)−𝜆₁𝑔²(𝑡)−𝑔²(0), (3.17) or 𝑔^′(𝑡)²≥2𝑐₁𝑔𝜎+1^𝜎+1(𝑡)−𝜆₁𝑔²(𝑡)−2𝑐₁𝑔𝜎+1^𝜎+1(0)+𝜆₁𝑔²(0)−𝑔²(0). (3.18) Denote 𝐴=𝑔^′(0)²+𝜆₁𝑔²(0)−2𝑐₁𝑔𝜎+1^𝜎+1(0). (3.19) If the condition (3.2) is fulfilled, that is, _Ω𝜔(𝑥)𝑢₀(𝑥)𝜓(𝑥)𝑑𝑥²+𝜆_1Ω𝜔(𝑥)𝑢₀(𝑥)𝜑(𝑥)𝑑𝑥²−2𝑐₁𝜎+1_Ω𝜔(𝑥)𝑢₀(𝑥)𝜑(𝑥)𝑑𝑥^𝜎𝜏+1>0 (3.20) and if 𝑔(0)>𝑐₂=(𝜆₁/2𝑐₁/(𝜎+1))^2/(𝜎+1), then lim_{𝑡→𝑇−0}𝑔(𝑡)=+∞. Hence we have lim_{𝑡→𝑇−0}max_Ω𝜔(𝑥)𝑢₀(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 Π₀, 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 Π₀.

Theorem 3.3. Let one assume that |𝑓(𝑥,𝑡,𝑢)|≤(𝑐₃+𝑐₄𝑡^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 Π₀ and |𝑢(𝑥,𝑡)|≤𝑐₅𝑒^−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 𝜆₁ of the boundary-value problem 𝐿_𝑝𝑢+𝜆₁𝑢=0,𝑥∈Ω,𝑢=0,𝑥∈𝜕Ω. (3.22) Let's consider the function 𝑉(𝑥,𝑡)=𝜀𝑒^{−𝜆1𝑡/2}𝜗(𝑥). We have 𝑉_𝑡𝑡−𝐿_𝑝()1𝑉−𝑓𝑥,𝑡,𝑉=𝜀4𝜆²₁+𝑐₂𝑒^{−𝜆1𝑡/2}𝜗(𝑥)−𝑐₃+𝑐₄𝑡^𝑚𝜀^𝜎𝑒^{−𝜆1𝑡/2}𝜗^𝜎()≥0,𝑥,𝑡∈Π₀,𝑉>0,(𝑥,𝑡)∈Γ₀, (3.23) 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≤𝑐₅𝑒^{−𝜆1𝑡/2}. Let us determine the class of functions |𝜑(𝑥)|≤𝛿,|𝜓(𝑥)|≤𝛿=𝜀⋅min_Ω𝜗(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^∞𝑅¹,𝜃(𝑡)≡1,if𝑡≤𝑇,𝜃(𝑡)=0,if𝑡>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_{𝑡→𝑇−0}max_Ω𝜔(𝑥)𝑢₀(𝑥)𝑢(𝑥,𝑡)=+∞, (3.25) where 𝑇=const>0.

Theorem 3.4. Let 𝑓(𝑥,𝑡,𝑢)≥𝑐_𝜎𝑒^{𝜆1𝜎𝑡}𝑢^𝜎 at (𝑥,𝑡)∈Π₀,𝑢≥0,𝜎=𝑐𝑜𝑛𝑠𝑡>1, and 𝜆₁ be positive eigenvalue of problem (3.16) in Ω which corresponds to the positive in Ω eigenfunction. If 𝑢(𝑥,0)≥0,𝑢^′_𝑡(𝑥,0)≥0,𝑢(𝑥,0)≠0, where 𝑢^′_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)≥−𝜆₁𝑔+𝐶_𝑔𝑒^𝜆1𝜎𝑔^𝜎(𝑡). (3.26)
Let 𝑔(𝑡)=𝜓(𝑡)𝑒^−𝜆1𝑡. From (3.25) it follows that 𝜓^′′≥𝑐₈𝜓^𝜏. Hence 𝜓(𝑡)→+∞ at 𝑡→𝑇−0. Thus it tends 𝑔(𝑡) to 𝑘+∞ and 𝑡→𝑇−0. Consequently, max_Ω(𝜔(𝑥)𝑢(𝑥,𝑡)) also tends to infinity.

Corollary 3.5. Let 𝑓(𝑥,𝑡,𝑢)≥𝑐₈𝑒^𝜆1𝜎𝑢^𝜎 at (𝑥,𝑡)∈Π₀,𝑢≥0,𝜎>1. Then there is no positive in Π₀ solution of the problem (2.1)–(2.3).
Let one consider the following equation: 𝜕²𝑢𝜕𝑡²=^𝑛_𝑖,𝑗=1𝜕𝜕𝑥_𝑖𝜔(𝑥)𝑢^𝜎𝜕𝑢𝜕𝑥_𝑗+𝑓(𝑥,𝑡,𝑢), (3.27) in bounded domain Ω∈𝑅^𝑛,𝑛≥2, and let 𝑢₀(𝑥)>0,∫_Ω𝑢₀(𝑥)𝑑𝑥=1, then one will have eigenfunction of the problem Δ𝑢+𝜆𝑢=0 in Ω,𝑢=0 on 𝜕Ω, and 𝜆₁>0 the corresponding eigenvalue.

Theorem 3.6. Let 𝑓(𝑥,𝑡,𝑢)≥𝛼₀|𝑢|^𝜎+1 at 𝛼₀=𝑐𝑜𝑛𝑠𝑡≥0 at (𝑥,𝑡)∈Π₀,𝑢≥0,𝜎>1. There exists 𝑘=𝑐𝑜𝑛𝑠𝑡>0 such that if 𝑢(𝑥,0)≥0,𝑢^′_𝑡(𝑥,0)≥0,∫_Ω𝑢(𝑥,0)𝑢₀𝑑𝑥≥𝑘,∫_Ω𝑢^′_𝑡(𝑥,0)𝑢₀𝑑𝑥≥𝑘 or condition (3.2) is fulfilled, then lim_𝑡→𝑇max_Ω𝜔(𝑥)𝑢₀(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 Π₀ and condition (2.2) is fulfilled on Γ₀. Substituting in corresponding integral identity Γ₀  𝜂=𝜀⁻¹𝑢₀(𝑥)𝜔(𝑥),𝑏=𝑎+𝜀,𝜀>0 to get 𝜀⁻¹_Ω𝜔(𝑥)𝑢_𝑡(𝑥,𝑎+𝜀)𝑢₀(𝑥)𝑑𝑥−_Ω𝜔(𝑥)𝑢_𝑡(𝑥,𝑎)𝑢₀(𝑥)𝑑𝑥+𝜀⁻¹_{Π𝑎,𝑎+𝜀}^𝑛_𝑖,𝑗=1𝜕𝜕𝑥_𝑖𝜔(𝑥)𝑢^𝜎𝜕𝑢𝜕𝑥_𝑗𝑢₀(𝑥)𝜔(𝑥)𝑑𝑥=𝜀⁻¹_{Π𝑎,𝑎+𝜀}𝑢₀𝜔𝑓(𝑥,𝑡,𝑢)𝑑𝑥𝑑𝑡, (3.29) where 𝑢₀(𝑥)>0. Tending 𝜀 to zero, we'll obtain 𝜕²𝜕𝑡²𝑔(𝑡)≥_Ω𝑢₀(𝑥)𝜔(𝑥)𝜕𝜕𝑥_𝑖𝜔𝑢^𝜎𝜕𝑢𝜕𝑥_𝑗𝑑𝑥+_Ω𝑢₀𝜔𝑢^𝜎+1𝑑𝑥, (3.30) where 𝑔(𝑡)=∫_Ω𝜔(𝑥)𝑢₀(𝑥)𝑢(𝑥,𝑡)𝑑𝑥. Let’s consider first the integral on the right-hand side in (3.30). Let 𝜑(𝑥)>0,𝜓(𝑥)>0 in Ω and sup𝑝𝜑(𝑥)⊂Ω,sup𝜓(𝑥)⊂Ω. Denote the surface of degeneration of (3.27), that is, the boundary of the solution sup𝑢(𝑡,𝑥)≡𝜃(𝑡). Since 𝑢(𝑡,𝑥)≡0 on 𝜕𝜃(𝑡) and Ω/𝜃(𝑡) from the Green's formula, we get _Ω𝜔𝑢₀𝜕𝜕𝑥_𝑖𝜔^𝜎₄𝜕𝑢𝜕𝑥_𝑗1𝑑𝑥≡𝜎+1_𝜃(𝑡)𝜔𝑢₀Δ𝑢^𝜎+1≡1𝑑𝑥𝜎+1_𝜃(𝑡)Δ𝑢₀(0)𝜔(𝑥)^𝜎+1𝑢𝑑_𝑥+_{𝜕𝜃(𝑡)}𝑢₀𝜔𝜕𝑢^𝜎+1𝜕𝑛_𝑡𝑑𝑠−_{𝜕𝜃(𝑡)}𝜕𝑢₀𝜔𝜕𝑛_𝑡𝑢^𝜎+1𝑑𝑥, (3.31) where 𝜕/𝜕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 _Ω𝜔𝑢₀(𝑥)𝜕𝜕𝑥_𝑖𝜔𝑢^𝜎𝜕𝑢𝜕𝑥_𝑗1𝑑𝑥=𝜎+1_Ω𝑢^𝜎+1Δ𝜔𝑢₀(𝑥)𝑑𝑥. (3.32) Using Δ𝑢₀(𝑥)𝜔(𝑥)=−𝜆₁𝑢₀(𝑥)𝜔(𝑥) we will get 𝜕²𝑔(𝑡)𝜕𝑡²≥𝜆1−¹𝜎+1_Ω𝑢^𝜎+1𝑢₀(𝑥)𝜔(𝑥)𝑑𝑥,𝑡>0. (3.33) Using the results of paper [12] we will obtain the required result.