1Uchinchi renessans yosh olimlari: zamonaviy vazifalar,
innovatsiya va istiqbol Young Scientists of the Third Renaissance: Current Challenges, Innovations and Prospects
NODIVERGENT TIPDAGI DIFFUZIYA TENGLAMASI UCHUN KOSHI MASALASINI TATQIQ QILISH
M. Aripov
f.-m. f.d., professor, O'zbekiston Milliy universiteti M. M. Boboqandov professorl, tayanch doktorant, Toshkent ijtimoiy innovatsiya universiteti katta o'qituvchi.
Ozbekiston.
q = x)| t > t0 = const > 0,x g RN,N > l| sohada quyidagi tenglama qaraymiz:
A I l"nl qj- (\ \n
Au = x ut - u divI x u Vu
Vu 1 + Ixl -1tlu ß = 0, (1)
u ( t, x )| t=% = uo ( x )> 0, x g Rn . (2)
q < 1,k > 0,m,ß> 1,p > 2,l,n,n berilgan o'zgarmaslar va u0(x) - nomanfiy,
notrivial boshlang'ich shart. (1) tenglama qator fizik - kimyoviy jarayonlarni ifodalaydi: bir jinsli bo'lmagan muhitlarda issiqlik tarqalish jarayonini, bir jinsli bo'lmagan muhitlardagi suyuqlik va gazning filtratsiyasi jarayonini, bir jinsli bo'lmagan muhitlarda reaksiya diffuziya va boshqa chiziqli bo'lmagan jarayonlarni ifodalaydi.
(1) tenglama xususiy hollarda bir qancha olimlar va tadqiqotchilar tomonidan o'rganilgan. Xususan, (1) tenglamani zichlik hadisiz manba holida, (1)-(2) masala yechimining sifat xossalari [4] muallif tomonidan tatqiq qilingan. [9] maqolada muallif bir jinsli, n = n = 0 va harakatlanuvchi muhitlarda (1)-(2) masala uchun global yechimlarning mavjudlik sharti yoki kritik Fujita ko'rsatkichini aniqlagan. (1) tenglamani divergent va sistema ko'rinishida q = 0, l = 0, n = 0 holida Matyakubov [8], Raxmonov va Alimov [10] o'z ilmiy ishlarida o'rgangan va Koshi yoki Neyman masalalari uchun global yechimlarning mavjudlik va mavjud bo'lmaslik shartlarini aniqlashgan. Martynenko va Tedeev [6] quyidagi tenglamalar uchun Koshi masalasini tatqiq etishgan:
p(x)ut = div(um-1 |Vu|2-1 Vu) + up,x g Rn,t > 0, (3)
va
p(x)ut = div(um-1 |Vu|2-1 Vu) + p(x)up, x g Rn , t >0. (4)
35
May 15, 2024
Uchinchi renessansyosh olimlari: zamonaviy vazifalar,
innovatsiya va istiqbol Young Scientists of the Third Renaissance: Current Challenges, Innovations and Prospects
Bu yerda 2>0,m + 2-2>0,p > m + 2-1,p(x) = \x\ " yoki p(x) = (1 + |x|) n.
Martynenko va Tedeev (3)-(4) tenglamada koeffitsientlar ma'lum bir shartlarni qanoatlantirganda (3)-(4) tenglamalar uchun Koshi masalasining yechimi chekli vaqtda cheksizlikka intilishini (blow-up) ko'rsatib o'tishgan [7].
Andreuchchi D.va Tedeev A.F. [1] quyidagi tengalama uchun Koshi masalasini yechim uchun quyi-yuqori chegaralar baholarini olishgan va yechimlarni fazoviy lokallashuvi ko'rsatishgan
U = div [um -1 \Du\2 -1 Du) + f (x) up,
bunda f - Lebeg o'lchovi ma'nosida chegaralanmagan yoki x ^ 0 da buziluvchan bo'lishi ham mumkin.
(1) tenglamaning divergent va nodivergent ko'rinishlarida q = n = n = l = 0,p = 2 va q = n = l = 0 hollarida Koshi masalasini J.Vazquez G'ovak muhitlar tenglamalari [11] kitobida Koshi masalasi yoki Dirak delta funksiyasi bilan berilgan Koshi masalasi yechimlarining mavjudlik sharti, mavjud
bo'lmaslik sharti, Lp va Wk,p fazolarda yechimlarning invariantligini o'rgangan. Ma'lumki, (1) tenglama buziluvchan [3]. Shu sababli, u = 0,Vu = 0 bo'lgan
sohalarda (1)-(2) masala klassik ma'noda yechimga ega bo'lmaydi. Shuning uchun bu
p-2,
hollarda 0„ u ( t, x
„ u(t,x),div||
\n m-1
x u
VuA
Vu jeC (Q) sinfga tegishli, (1) tenglamani
integral taqsimot ma'nosida qanoatlantiruvchi [2] umumlashgan yechimlarni qaraymiz.
(1) tengalamada v = u1 - belgilash kiritamiz va Au operator quyidagicha ko'rinishga keladi:
Au = Ixl ni v - div
I in m-i
x v 2
\p-2
Vv2 Vv 1 + (1- q )| x|" nit
- n A A _
v
0,
(5)
v It=tn = v0 (x) = (u0 (x))
1-q
(6)
^ , m- q, ß-q Bu yerda: m2 =-, k2 =-,ß2 =-
1-q
1-q 1-q
Q sohaga /2 normani kiritamiz va r = |x|; bo'lsin. U holda, (6) tenglamani
quyidagicha yozib olamiz:
Au = r""1 -r1" ^
..n+N-
-1 m--1U k- \ v2 (v2 )r
p-2 ^ v
+ (1- q ) r-1 = 0
(7)
May 15, 2024
36
Uchinchi renessans yosh olimlari: zamonaviy vazifalar,
innovatsiya va istiqbol Young Scientists of the Third Renaissance: Current Challenges, Innovations and Prospects
(7) tenglamaning yechimini quyidagicha ko'rinishda qidiramiz:
v ( t, r ) = t ~af (Ç) ,Ç = rt-Л (8)
Bu yerda
( nx +1)( p - n - nx ) ß -1 -( Щ +1)( m2 + k2 ( p - 2 )-1)
a =
,/л
d d d = (ß-1)( p - n - n 0.
(8) belgilashni (7) tenglikka kiritamiz va (7) tenglama quyidagi k o'rinishga kelishini ko'rish qiyin emas
p-2 Л
И-N d
Ç dÇ
^N-1+n j-m2 -1
df2
dÇ
f +Ç1 f + aÇ~"lf -(1 - q)Г/ß2=0. )
(9)
Boshlang'ich shartga muvofiq, (9) tenglamaning notrivial, nomanfiy va quyidagi shartlarini qanoatlantiruvchi yechimlarini qidiramiz:
f '(0) =f (b) = 0,0 < b < +«. (10) Quyidagicha belgilash kiritamiz:
z(x,t) = t~af (Ç),f (Ç) = A(Ç - Çy f+2 (11)
Bu yerda:
p
У =-7 У
p -1
p - 1 m2 + k2(p - 2) - 1
, A ■■
p -1
1
pÏ2
1 Y2
p-1
V pk2~- ) V )
, (b) = max (b,0) , Ç0 = const > 0
Teorema. v(t,x) (3)-(4) masalaning umumlashgan yechimi bo'lsin va
quyidagi
tengsizliklar
p - n - n
va
y2 > 0, p > n + nx,ß2 >ß2C = 1 + (1 +1 ) m2 + k2 (p - 2)-1 +
V N - n
v(x,t0) < z(x,t0),x eRn, o'rinli bo'lsin. U holda, Q sohada (3)-(4) masalaning yechimi uchun quyidagi baho o'rinli:
v(t,x)„ z(t,x).
Isbot. Teoremani isbotlash uchun taqqoslash prinsipini qo'llaymiz [2] va z (t, x) funksiyani taqqoslovchi funksiya sifatida tanlaymiz. (11) belgilashni (3) tenglikka kiritamiz va quyidagi tenglikka ega bo'lamiz:
May 15, 2024
37
Uchinchi renessansyosh olimlari: zamonaviy vazifalar,
innovatsiya va istiqbol Young Scientists of the Third Renaissance: Current
ChaL^ndR^andrr^
f
H- N d
g dg
^N-l+nym2-1
df
dg
p-2
f dg
^f"1 f + af1f-(1-q)fnif"2 < 0. (12)
dg
p-2
_ d f (g) funksiyani —
^N-1+«ym2-1
df'
dg
df_
dg
+ (gN "f) = 0 tenglamani yechimi
ekanligi sababli, (12) tengsizlikni quyidagicha yozib olamiz:
-f-'-"1/((1-q) f +M( N-,h)-o)< 0. (13)
g, f > 0 va y2> 0, p > " + " ,fi2> P2c yoki N-" )<a ekanliginidan (13) tengsizlik har doim bajariladi. U holda, z(t0,x)>v0(x), x^uN, bo'lsa, fl sohada v (t, x) < z (t, x) tengsizlik o'rinli. Teorema isbotlandi.
REFERENCES
1. Andreucci D. , Tedeev A. F. Universal bounds at the blow-up time for nonlinear parabolic equations. Advances in Differential Equations, 10(1), 2005, pp. 89-120.
2. Aripov M. and Sadullaeva S. Computer simulation of nonlinear diffusion processes, National University of Uzbekistan Press, 2020.
3. Aripov M., Bobokandov M. The Cauchy Problem for a Doubly Nonlinear Parabolic Equation with Variable Density and Nonlinear Time-Dependent Absorption, Journal of Mathematical Sciences, 2023, 277(3), pp. 355-365.
4. Bobokandov, Makhmud. Investigation of weak solutions of double nonlinear parabolic equation in non-divergence form with source term. Scientific Results 3 (2023).
5. Galaktionov V.A. and Levine H.A. A general approach to critical Fujita exponents and systems. Nonlinear Anal., 34, 1998, pp. 1005-1027.
6. Martynenko A. V. and Tedeev A. F. The Cauchy problem for a quasilinear parabolic equation with a source and nonhomogeneous density. Comput. Math. Math. Phys., 47, 2007, pp. 238-248.
7. Martynenko A. V., Tedeev A. F. and Shramenko V.N. Cauchy problem for a degenerate parabolic equation with inhomogeneous density and source in class slowly tending to zero initial functions. Izv. RAN. Ser. Mat., 76, 2012, pp. 139-156.
8. Matyakubov, A. S. Finite speed of the perturbation distribution and asymptotic behavior of the solutions of a parabolic system not in divergence form. Univ. J. Comput. Math 5 (2017): 57-67.
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May15,2024
Uchinchi renessans yosh olimlari: zamonaviy vazifalar,
innovatsiya va istiqbol Young Scientists of the Third Renaissance: Current Challenges, Innovations and Prospects
9. Mersaid, Aripov. The fujita and secondary type critical exponents in nonlinear parabolic equations and systems. In Differential Equations and Dynamical Systems: 2 USUZCAMP, Urgench, Uzbekistan, August 8-12, 2017, pp. 9-23. Springer International Publishing, 2018.
10.Rakhmonov, Z. R., and A. A. Alimov. Properties of solutions for a nonlinear diffusion problem with a gradient nonlinearity. International Journal of Applied Mathematics 36, no. 3 (2023): 405.
11.Vazquez J.L. The porous medium equation. Oxford, Clarendon press, 2007.
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