To The Qualitative Properties Of Solution Of System Equations Not In Divergence Form

To the Qualitative Properties of Solution of System

Equations not in Divergence Form

Mirsaid Aripov1,Alisher Matyakubov2

1,2_{Applied Mathematics and Computer Analysis, National University of Uzbekistan,}

Tashkent, 100174, Uzbekistan

Abstract In this paper the properties of solutions of nonlinear systems of parabolic equations not in diver-gence form

|x|n ∂u_∂t =uγ1∇

(

|∇u|p−2∇u

)

+ |x|nvq1_,

|x|n ∂v ∂t =v

γ2∇

(

|∇v|p−2∇v

)

+ |x|n_uq2_,

are studied. In this work used: method of nonlinear splitting, known previously for non-linear parabolic equations and systems of equations in divergence form, asymptotic theory and asymptotic methods based on different transformations. Constructed asymptotic representation of self-similar solutions of nonlinear parabolic systems of equations not in divergence form, depending on the value in the system of the numerical parameters necessary and sufficient signs of their existence. The main purpose of this paper is to find conditions for the existence and non-existence results for global solutions of parabolic equations not in divergence form on the basis of the self-similar analysis.

Keywords nonlinear parabolic systems of equations, not in divergence form, global solutions, self-similar solutions, asymptotic representation of solution

1

Introduction

Consider in Q={(t, x) : t >0, x∈RN}parabolic system of two quasilinear equations not in divergence form

|x|n∂u ∂t =u

γ1∇

(

|∇u|p−2∇u

)

+|x|nvq1_,

|x|n∂v ∂t =v

γ2∇

(

|∇v|p−2∇v

)

+|x|nuq2_,

(1)

u|t=0=u0(x)≥0, v|t=0=v0(x)≥0, ∀x∈RN (2)

where n, p, γi, qi (i = 1,2)− the numerical parame-ters set,∇(·) =gradx(·), tandx∈RN_{−respectively,}

the temporal and spatial coordinates, u=u(t, x)≥0, v=v(t, x)≥0−are the solutions.

The numerical parameterncharacterizes the variable density of the nonlinear medium. The system of equa-tions (1) describes the process of polytrophic filtration

in a nonlinear two-componential medium with variable density. The system of equations (1) is called the system of equations of polytrophic filtration, two-componential nonlinear medium.

In this system u ≥ 0, v ≥ 0− means the pres-sure,|∇u|p−2∇u,|∇v|p−2∇v−filtration flow,uq2_{, v}q1−

power volume filtration sources.

The system of equations (1) describes many physical phenomena [2–10]. In particular, atγi= 2, p= 2, qi = 3, n= 0 for a single equation in (1) it is encountered in plasma physics [2]. This problem, whenN = 1,arises in a model for the resistive diffusion of a force-free magnetic field in a plasma confined between two walls {0 < z <

∞}.The magnetic field has the formB0(cosϕ,sinϕ,0) withB0 constant andϕ=ϕ(z, t).

In [3] Zhou and Yao are studied the Cauchy problem (1)-(2) for p= 2, n= 0 and the absence of absorption, proved the existence of a single viscous solutions, and in [4] Wang is investigated the existence and uniqueness of a classical solution of the Cauchy problem for p = 2, n= 0.

In [5] Wang and Wei are considered a de-generate nonlinear parabolic system with localized source ut = uα_(∆u+up_{(x, t)v}q_(x

0, t)), vt = vβ_(∆v+vm_{(x, t)u}n_(x

0, t)). In [5] deals with blow-up properties for a degenerate parabolic system with nonlinear localized sources subject to the homogeneous Dirichlet boundary conditions. The main aim of [5] is to study the up rate estimate and the uniform up profile of the up solution. At the end, the blow-up set and blow blow-up rate with respect to the radial vari-able is considered when the domainQis a ball.

In [6] Zhi and Li studied the nonlinear degen-erate parabolic system ut = vγ1(uxx+au), vt = uγ2_(v

xx+bv) with Dirichlet boundary condition . The

regularization method and upper-lower solutions tech-nique are employed to show the local existence of a solu-tion for the nonlinear degenerate parabolic system. The global existence of a solution is discussed. The finite time blow-up result together with an estimate of the blow-up time are found. The blow-up set with positive measure is analyzed in some detail.

form:

uit=fi(ui+1) (∆ui+aiui), i= 1,2, ..., n−1, unt=fn(u1) (∆un+anun), x∈Ω, t >0

with homogenous Dirichlet boundary condition and pos-itive initial condition, where ai≥0 (i= 1,2, ..., n) and

fi(i = 1,2, ..., n) satisfy to some conditions are stud-ied. The local existence and uniqueness of classical so-lution are proved. Moreover, it is proved that: (i) when min{a1, ..., an} ≤ λ1 then there exists global positive classical solution, and all positive classical solutions can not blow up in finite time in the meaning of maximum norm; (ii) when min{a1, ..., an}> λ1 and the initial da-tum (u10, u20, ..., un0) satisfies some assumptions, then the positive classical solution is unique and blows up in finite time, whereλ1 is the first eigenvalue of−∆ in Ω with homogeneous Dirichlet boundary condition.

In [8], Chunhua and Jingxue are concerned with the self-similar solutions of the form

u(t, x) = (t+ 1)−αf

(

(t+ 1)β|x|2

)

for the following degenerate and singular parabolic equa-tion in non-divergence form

∂u ∂t =u

m_div(_|∇u|p−2_∇ u

)

, m≥1, p >1.

First established the existence and uniqueness of solu-tionsf with compact supports, which implies that the self-similar solution is shrink. On the basis of this, also established the convergent rates of these solutions on the boundary of the supports. On the other hands, also con-sidered the convergent speeds of solutions, and compare which with Dirac function as t tends to infinity.

In [9], Raimbekov studied some properties of the so-lutions of the Cauchy problem for a nonlinear parabolic equations in non-divergence form with variable den-sity |x|n ∂u_∂t = um_div(_|∇u|p−2_∇

u

)

, p > 1, 0 ≤

m < (p−2)(N_p−+_Nn)+p+n received self-similar solution Barenblatt-Zeldovich-Kompaneets type and methods of the theory of comparisons prove the asymptotic behav-ior of solutions in the case of fast and slow diffusion. This article also gives some comparative numerical re-sults for the casem= 0, m= 1 andm= 1,5.Using this result the author talks about the properties of the finite speed of propagation of heat for divergent equations and localization for non-divergent case.

Aripov and Matyakubov [10] studied the asymptotic behavior of self-similar solutions of a parabolic equation (1) for the case n = 0. Constructed asymptotic repre-sentation of self-similar solutions of nonlinear parabolic systems of equations not in divergence form, depending on the value in the system of the numerical parameters necessary and sufficient signs of their existence.

In this paper using self-similar approach we find a par-ticular solution of the system (1), and it is proved this solution asymptotic of compactly supported solutions. The main purpose of this paper is to find conditions for the existence and non-existence results for global solu-tions of problem (1)-(2) on the basis of the self-similar analysis [1,12].

2

Self-similar

system

of

equa-tions (1)

Transform the system (1) to the relatively easy to study mind. To receive this auxiliary system of equa-tions is applicable to a system (1) the following trans-formation

u(x, t) = (t+T)−α1_f(ξ),

v(x, t) = (t+T)−α2_φ(ξ),

ξ= (t+T)−γ|x|,

(3)

where α1=−₁1+_−qq₁1_q2, α2=−₁1+_−q1q2_q2, T >0, nγ = 1 + (1+q1)(p+γ1−2)

1−q1q2 , α1(p+γ1−2) =

α2(p+γ2−2) ,we get the self-similar system of equa-tions

fγ1_ξ1−N d

dξ

(

ξN−1df dξ

p−2df

dξ

)

+α1ξnf+

+γξn+1df dξ +ξ

n_φq1_{= 0,}

φγ2_ξ1−N d

dξ

(

ξN−1dφ dξ

p−2dφ_dξ

)

+α2ξnφ+

+γξn+1dφ dξ +ξ

n_fq2 _{= 0.}

(4)

The case n = 0 was considered in [10]. In the work [11], the qualitative properties of solutions of system (4) in divergence form are studied based on the self-similar and approximately self-similar approach, one way of con-struction of the critical exponent and property finite speed of perturbation (FSP) for system (1) are estab-lished.

3

Slowly diffusion case:

p

+

γ

−

2

>

0

, i

= 1

,

2

.

A global solvability

of solutions

We prove properties of a global solvability of weak so-lutions of the system (1) using a comparison principle [13]. For this goal, we construct a new system of equa-tion using the standard equaequa-tion method as in [1,12]:

u+(t, x) = (t+T)−

α1_f(ξ),

v+(t, x) = (t+T)−α2φ(ξ),

(5)

where α1= _q1+_1q2q₋1₁, α2=_q1+_1q2q₋2₁, T >0, ξ= (t+T)−γ|x|, nγ= 1 +α1(p+γ1−2). In the case, α1(p+γ1−2) =α2(p+γ2−2)

f(ξ) =A1

(

a−ξpp−+n1

)

+

p−1

p+γ1−2

,

φ(ξ) =A2

(

a−ξpp−+n1

)

+

p−1

p+γ2−2

,

(6)

where a >0, Ai=

(

γ(p+γi−2)p−1

(1−γi)(p+n)p−1 ) 1

p+γi−2

, i= 1,2, b+= max (0, b).

ki=

(p₋1)qi

p+γ3−i−2− p−1

p+γi−2, hi=

p+n+(1₋n)γi−2

n(1−γi) ,

mi=A−_i 1Aqi

3−i, i= 1,2.

Theorem 1. Letp+γi−2>0, qi> p+γ3−i−2 p+γi−2 ,

− N+n

n(1−γi)+

(1 +qi)hi q1q2−1

+miaki ≤0, i= 1,2,

u+(0, x)≥u0(x), v+(0, x)≥v0(x), x∈RN.

Then for sufficiently small u0(x), v0(x) the followings holds

u(t, x)≤u+(t, x), v(t, x)≤v+(t, x) in Q, (7)

where the functionsu+(t, x), v+(t, x) defined as above. Proof. Theorem 1 is proved by the method of com-parison of solutions. As a comcom-parison solution we take the functionsu+(t, x), v+(t, x).Substituting (5) in (1) we obtain the following inequality

fγ1ξ1−N d dξ

(

ξN−1df dξ

p−2df

dξ

)

+α1ξnf+

+γξn+1df dξ +ξ

n_φq1 ≤_0,

φγ2_ξ1−N d

dξ

(

ξN−1dφ dξ

p−2dφ_dξ

)

+α2ξnφ+

+γξn+1dφ dξ +ξ

n

fq2≤0.

(8)

Given the specific form (6) of the functions

f(ξ), φ(ξ) inequality (8) can be rewritten as follows:

− N+n

n(1−γ1)

+(1 +q1)h1

q1q2−1 +m1

(

a−ξpp−+n1

)k1

≤0,

− N+n

n(1−γ2)

+(1 +q2)h2

q1q2−1 +m2

(

a−ξpp−+n1

)k2

≤0.

It is easy to check that m1

(

a−ξpp−+n1

)k1

≤ m1ak1,

m2

(

a−ξpp−+n1

)k2

≤m2ak2.

Then according to the hypotheses of Theorem 1 and comparison principle we have

u(t, x)≤u+(t, x), v(t, x)≤v+(t, x) in Q, ifu+(0, x)≥u0(x), v+(0, x)≥v0(x), x∈RN. The proof of the theorem is complete.

4

Asymptotic of the self-similar

solutions

Next, we study the asymptotic behavior of the self-similar solutions of the system (4). Self-self-similar solution of system equations (4) will search for in the form

f(ξ) =f(ξ)y(η), φ(ξ) =φ(ξ)z(η), (9)

where η =−ln

(

a−ξpp−+n1

)

, f(ξ) =

(

a−ξpp−+n1

) p−1

p+γ1−2

,

φ(ξ) =

(

a−ξpp−+n1

) p−1

p+γ2−2

, a >0.

Then substituting (9) into (4) for the functiony(η)>

0, z(η) > 0 we have the following nonlinear system of equations

yγ1 d

dη(L1y) +a11(η)y γ1_(L

1y) +a13(η)zq1+

+a12(η)

(

dy

dη +a10(η)y

)

+a14(η)y= 0,

zγ2 d

dη(L2z) +a21(η)z γ2_(L

2z) +a23(η)yq2+

+a22(η)

(

dz

dη +a20(η)z

)

+a24(η)z= 0. (10)

Here ai0(η) =−p+pγ−i1−2, ai2(η) =γ (

p−1 p+n

)p−1 ,

ai1(η) =

(N+n)(p₋1) p+n

e−η a−e−η −

(p₋1)(1₋γi)

p+γi−2 ,

ai4(η) =αi

(

p−1 p+n

)p

, ai3(η) =

(

p−1 p+n

)p e−siη a−e−η , si = 1 + (p−1)qi

p+γ3−i−2− p−1

p+γi−2 (i= 1,2).

L1y=dy_dη +a10(η)y p₋2(

dy

dη +a10(η)y

)

,

L2z=dz_dη+a20(η)z p₋2(

dz

dη+a20(η)z

)

.

There was supposed to ξ ∈ [ξ0, ξ1), 0 < ξ0 < ξ1, ξ1=a

p−1

p+n_.

Therefore, the functionη(ξ) has properties: η′(ξ)>0 at ξ∈[ξ0, ξ1), η0=η(ξ0)>0,limξ0→ξ1η(ξ) = +∞.

Further, in what follows the auxiliary system of equa-tions (7) is investigated in the following limited:

lim

η→+∞aij(η) =a 0

ij (i = 1,2; j = 0,1,2,3,4)

are exists, finite and nonzero, those 0<a0

ij<+∞.

Through the introduction of transformations (3), (9) and properties η → +∞, study of the solutions of (1) reduced to the study of the solutions of (10), each of which is in a neighborhood +∞satisfies the inequalities

y(η)>0,y′+a10(η)y̸= 0,

z(η)>0,z′+a20(η)z̸= 0.

We now study the asymptotic behavior of positive, have nonzero a finite limit as η→+∞solutions of (10).

5

The main results

We introduce the notations:

ci1 = _(p+1_γ−γi

i−2)p, ci2 =

1 (p+n)p−1

(

αi

p+n− γ p+γi−2

)

, ci3= _(p+1_n)p_a ,(i= 1,2).

Let y(η) = y0+o(1), z(η) =z0+o(1) at η →+∞ and is performed the equality (1 +q1) (γ1+p−2) = (1 +q2) (γ2+p−2).

Then are valid the following theorem:

self-similar solution of equation (1) has the asymptotic

uA(t, x) =y0(T+t)

1+q1 1−q1q2×

×

(

a−

(

|x|

(t+T)γ

)p+n p−1

) p−1

p+γ1−2

+

(1 +o(1)),

vA(t, x) =z0(T+t)

1+q2 1−q1q2×

×

(

a−

(

|x|

(t+T)γ

)p+n p−1

) p−1

p+γ2−2

+

(1 +o(1)),

(11)

at |x| →app−+n1_(t+T)γ_{, where 0< y}0 _<+∞_{, 0< z}0_<

+∞andy0_{, z}0_{are the respectively rootsw}

1, w2the sys-tem of nonlinear algebraic equations

ci1wip+γi−1+ci2wi+ci3wq3i₋i = 0 (i= 1,2). (12)

Theorem 3. Let s1 = 0, s2 > 0. Then the self-similar solution of equation (1) has the asymptotic at |x| → app−+n1_(t+T)γ _{form (11), where 0 < y}0 _{< +}∞_,

0 < z0 _{< +∞ and y}0_{, z}0 _{are the respectively roots} w1, w2the system of nonlinear algebraic equations

c11w p+γ1−1

1 +c12w1+c13w q1

2 = 0, c21w2p+γ2−1+c22w2 = 0.

Theorem 4. Let s1 > 0, s2 = 0. Then the self-similar solution of equation (1) has the asymptotic at |x| → app−+n1_(t+T)γ _{form (11), where 0 < y}0 _{< +}∞_,

0 < z0 < +∞ and y0, z0 are the respectively roots

w1, w2the system of nonlinear algebraic equations

c11w p+γ1−2

1 +c12w1 = 0, c21w

p+γ2−2

2 +c22w2+c23w q2

1 = 0.

Theorem 5. Let s1 > 0, s2 > 0. Then the self-similar solution of equation (1) has the asymptotic at |x| → app−+n1_(t+T)γ _{form (11), where 0 < y}0 _{< +}∞_,

0 < z0 < +∞ and y0, z0 are the respectively roots

w1, w2the system of nonlinear algebraic equations

c11w1p+γ1−2+c12w1 = 0, c21w

p+γ2−2

2 +c22w2 = 0.

The proof. Assuming that the system (10)

ϑ1(η) =L1y, ϑ2(η) =L2z (13)

obtain the identity

ϑ′₁(η)≡ −a11(η)ϑ1(η)−a12(η)y−γ1ϑ

1

p−1

1 (η)− −a13(η)y−γ1zq1 − a14(η)y1−γ1,

ϑ′₂(η)≡ −a21(η)ϑ2(η)−a22(η)z−γ2ϑ

1

p−1

2 (η)− −a23(η)z−γ2yq2 − a24(η)z1−γ2.

(14)

Now consider the function

g1(λ1, η)≡ −a11(η)λ1−a12(η)y−γ1λ1

1

p−1−

−a13(η)y−γ1zq1 −a14(η)y1−γ1, g2(λ2, η)≡ −a21(η)λ2−a22(η)z−γ2λ2

1

p−1−

−a23(η)z−γ2yq2 − a24(η)z1−γ2.

(15)

where λi ∈R, (i= 1,2).

Suppose first si = 0 (i = 1,2). Then the

func-tions gi(λi, η) (i = 1,2) preserves a sign on an interval

[η1,+∞)⊂[η0,+∞) for each fixed valueλi (i= 1,2),

different from the values satisfying the system

−a011λ1−a012

(

y0)−γ1λ1

1

p−1 −_a0

13

(

y0)−γ1(z0)q1−

−a014

(

y0)1−γ1 = 0,

−a021λ2−a022

(

z0)−γ2λ2

1

p−1 −_a0

23

(

z0)−γ2(y0)q2−

−a0₂₄(z0)1−γ2= 0.

Now let si > 0 (i = 1,2). It is easy to see that the

functions gi(λi, η) (i= 1,2) for each fixed valueλi (i=

1,2),different from the values satisfying the system

−a0₁₁λ1−a012

(

y0)−γ1λ1

1

p−1 − _a0

14

(

y0)1−γ1 = 0,

−a0

21λ2−a022

(

z0)−γ2

λ2

1

p−1 −_a0

24

(

z0)1−γ2

= 0.

preserves a sign on an interval [η2,+∞)⊂[η0, +∞). And in the case si < 0 (i = 1,2) the functions gi(λi, η) (i= 1,2) rewritten in the following form

g1(λ1, η)≡ −a11(η)λ1−a12(η)y−γ1λ1

1

p−1−

−a13(η)y1−γ1

(

y−1_zq1 −_a

14(η)a−131(η)

)

, g2(λ2, η)≡ −a21(η)λ2−a22(η)z−γ2λ2

1

p−1−

−a23(η)z1−γ2

(

z−1yq2 − _a

24(η)a−231(η)

)

.

From here mean lim

η_→+_∞ ai1(η) = −

(p−1)(1−γi)

p+γi−2 ,

lim

η_→+_∞ ai2(η) = γ

(

p−1 p+n

)p−1 , lim

η_→+_∞ ai3(η) = ∞, lim

η_→+_∞ ai4(η) = αi

(

p−1 p+n

)p

(i= 1,2) implies that the

functions gi(λi, η) (i = 1,2) preserve sign on the in-terval [η2,+∞) ⊂ [η0,+∞), where λi ̸= 0 (i = 1,2).

That means the functions gi(λi, η) (i = 1,2) for all η ∈[ηi,+∞) (i= 1,2) satisfy one of the inequalities

gi(λi, η)>0 or gi(λi, η)<0 (i= 1,2). (16)

Suppose now that for the functions ϑi(η) (i = 1,2)

limit as η → +∞ does not exists. Consider the case where executed any of the inequalities (16). In view the oscillation of the functions ϑi(η) (i= 1,2) straight line

ϑi =λi (i= 1,2) its graph intersects the infinite num-ber of times in the interval [ηi,+∞) (i= 1,2).But this is impossible, since on the interval [ηi,+∞) (i= 1,2) rightly one of the inequalities (16) and, therefore, the identity (15) follows, that the graph of the functions

ϑi(η) (i= 1,2) intersects the straight linesϑi=λi (i= 1,2) only once on the interval [ηi,+∞) (i= 1,2). There-fore, for the functions ϑi(η) (i= 1,2) are exists limit at

η →+∞.

By assumption,y(η) =y0_{+o(1), z(η) =z}0_{+o(1) at} η →+∞,and the functionsϑi(η) (i= 1,2) identified in

accordance with (13) and has a limit atη →+∞.Then

y′(η) and z′(η) has a limit atη →+∞, and is equal to zero.

Then

ϑ1(η) =dy_dη +a10(η)y p₋2(

dy

dη +a10(η)y

)

=

=a0₁₀y0p−2a0₁₀y0+o(1), ϑ2(η) =dz_dη +a20(η)z

p₋2( dz

dη +a20(η)z

)

=

at η → +∞ and by (14) derivative of functions

ϑi(η) (i= 1,2) has a limit atη →+∞, which is

obvi-ously equal to zero.

Therefore, it is necessary in order to

lim

η_→+_∞

(

a11(η)ϑ1(η) +a12(η)y−γ1ϑ

1

p−1

1 (η)

)

+

+ lim

η→+∞

(

a13(η)y−γ1zq1 + a14(η)y1−γ1

)

= 0,

lim

η_→+_∞

(

a21(η)ϑ2(η) +a22(η)z−γ2ϑ

1

p−1

2 (η)

)

+

+ lim

η→+∞

(

a23(η)z−γ2yq2 +a24(η)z1−γ2

)

= 0.

From here easy to be convinced in the fact that at

si < 0 (i = 1,2) system (13) can not have solutions

(y(η), z(η)) with a finite non-zero limit, at η → +∞,

and at si ≥0 (i = 1,2) for the existence of such solu-tions is necessary, order to comply with the condisolu-tions of the theorem 2,3,4,5.

Consequently, by the transformations introduced by (3) and (9), self-similar solution of the system equation (1) has an asymptotic at |x| → app−+n1(t+T)γ and has

the following form

uA(t, x)≃y0(T+t)

1+q1 1−q1q2

(

a−

(

|x|

(t+T)γ

)p+n p−1

) p−1

p+γ₁−2

+

vA(t, x)≃z0(T+t)

1+q2 1−q1q2

(

a−

(

|x|

(t+T)γ

)p+n p−1

) p−1

p+γ₂−2

+ .

The theorems are proved.

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