Luận văn thạc sĩ -ngành Toán Giải Tích-Chuyên đề :Khảo sát phương trình sóng phi tuyến trong không gian Sobolev có trọng
Trang 1CHUaNG 1
PHAN TONG QUAN
Trang 1u~nvan nay, chung toi nghien cUu st! t6n tC;liva duy nhclt nghit%m
cua phuong trlnh s6ng phi tuyC'ntrang mQtilia troll don vi thuQcdC;lng
u"-(u" +~u,)+ F(u,u,)= I(r,t), 0 < r <1,0 <t <T,
1
1imJ;.Ur(r,t)
1
<+00, -ur(l,t) = hou(l,t) + h(t) urCl,t) + get),
r->O+
(1.1)
(1.2)
I l
a-l
P-l
F(u,ur)=F;(u) +F2(ur)= u u + Ur up (1.4)
trang d6 0 < a, j3< 3, ho> 0 la ba hfing s6 cho truoc, uo' Ul' f, g, h 1a cac ham
s6 cho truoc thoa mQt s6 di€u kit%nse duQc chi ra sail
Phuong trlnh (1.1) mo ta dao dQng phi tuyC'n cua mang troll don vi
{(x,y): r = ~X2+ y2 < I}, trang d6
~ u(r,t) 1a dQdich chuy€n theo phuong vuong g6c so voi vi to din bfing tC;li
O<r<l,O<t<:T.
~ f(r,t)-F(u,uJ 1a ngoC;li 1t!c, ~(u)=lula-lu 1a 1t!c can dan h6i,
F2(ur)= IUJH Urla 1t!ccan nhot
mang tC;limQi di€m tren bien r= 1 Di€u kit%nI!i~Frur (r, t)1< +00 se tt! dQng
thoa nC'u u 1a nghit%mc6 di€n, ch~ng hC;lnuEC2((O,1)x(O,T))nC1([O,1]x[O,T]).
Sobo1ev c6 trQng va chuy€n d6i bai toan v€ dC;lngbiC'n phan
Trang 2>- Di~u ki~n (1.3) ma tadQ l~ch ban d~u va v~n to'c ban d~u cua mQi di€m .
tren duong troll Cr,
Vdi cac di~u ki~n bien -ur(l,t) = hou(1,t), ho > 0, ho~c -ur(l,t) = h(t)u(1,t) + g(t), h(t)"2.ho > 0 nhi~u tac gia g~n day dii khao sat cho phuong trlnh song co
d<;lng
U/t-a(t)( Urr+~Ur) = f(r,t,u,ur,Ut)' 0 < r < 1,0 < t < T,
ch£ng h<;lnnhu
( 1.5)
[3] (Binh, 1997), vdi
f = J;(r,t)- F(u),
I l
a-l
f = - Ut Up 0 < a < 1, aU) =:1.
[7] (Long, Dinh, Binh, 1999), vdi
f=J;(r,t)-F(u),
I I
P-2
f=J;(r,t)- u u,l<p<3,
f=J;(r,t)-luJ sign(ut), O<a<l, aU)=: 1.
[2] (Binh, Dinh, Long, 2002), vdi aCt)=: 1,
f = f(r,t,u,ur)'
f=f(u).
Truong hQp khang co sO'h<;lng !ur dIng duQc nhi~u tac gia khao sat, ch£ng
r
h<;ln
f=-Iulasign(u), O<a<l,
f=-Iutlasign(ut), O<a<1.
[6] (Long, Dinh, 1995), vdi aCt)=: 1,
f = f(u,uJ.
Trang 3Nhu' v~y phu'dng trlnh (1.1) voi sO'h~ng phi tuye'n F(U,ut) cho d (1.4) la
mOt tru'ang h<jprieng cua phu'dng trlnh (1.5), nhu'ng lien ke't voi di~u ki<%nbien han h<jpkhong thucinnha't (1.2) ma tru'oc day chu'a du'<jckhao sat
Trang lu~n van nay, chung toi khao sat slf t6n t~i va duy nha't nghi<%mcua
bai toan (1.1)- (1.4) voi hai nhom gia thie't khac nhau Cong Cl:ldu'<jcstt dl:lng
d€ nghien CUuslf t6n t~i va duy nha't nghi<%mla phu'dng phap Galerkin, dinh ly
di€m ba't dOng Schauder, phu'dng phap compact yell, phu'dng phap toah ttt ddn di<%utrang khong gian Sobolev co trQng thich h<jp
Toan bO lu~n van du'<jc chia thanh 4 chu'dng, phcin ke't lu~n va tai li<%u tham khao
y Chu'dng 1: GiOi thi<%ut6ng quat v~ bai toan va di€m qua cac ke't qua tru'oc
do, d6ng thai giOithi<%utom Hitcac chu'dngtie'p theo
y Chu'dng 2: Trlnh bay cac khong gian ham Sobolev co trQngva cac tinh cha't v~ cac phep nhung compact gifi'a cac khong gian ham; cac khong gian ham
LP(0, T; X), 1 ~ P ~ 00; mOt sO' ba't d~ng thilc va cac cong Cl:lgiai tich khac co
lien quail de'n cac chu'dng sail
-( 1.4) vOi nhom gia thie't thil nha't
y Chu'dng 4: Nghien CUuslf t6n t~i va duy nha't nghi<%mcua bai toan (1.1) -(1.4) voi nhom gia thie't thil hai
y Ph~n ke't lu~n
y Tai li~u tham khao