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p
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p
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p
p
Qnp
p
M
p
p
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L
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p
p
p p
p n
p
p
Qp p Zp p pZp Zp Zp/pZp Fp Qp x Qp v v(p) = 1 |x|p = p
v(x)
ac(x) =xpv(x) Qp Zp p1Zp
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S(Qnp ) C Qnp
S(Qnp ) Qnp x y Q
np x y=
ni=1 xiyi
f =f(x) Qp
[x1
, . . . , xn
]
p f(, ) |f|p
f(, )= F1
|f|pF
,
F :S(Qnp )S(Qnp )
Qnp
(x y)(x) dnx
Qnp dnx Qnp
Zn
p
1 f(, ) L2(Qnp ) f(, ) p
f(, )u= g;
E u= E g
Du
(x) =F1x
||pFxu
, >0, u S(Qp).
p 1
2i
dudx
=
R
e2ix
Fxu
() d, N.
p
u(x, t)
t + a
0(x, t)(Du)(x, t) +
n
k=1 ak(x, t)(Dku)(x, t)+ b(x, t)u(x, t) =f(x, t), x Qp, 0< t T ,
u(x, 0) =(x),
0< 1 < < n < D
ak(x, t)f(x, t)
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ak(x, t) k= 0, 1, . . . , n b(x, t)
t
M (x) Qp
|(x)| C(1 + |x|p);
t M t C t
0 < 1 M t
u(x, t) =
t0
Qp
(x,t,, )f(, ) d d+
Qp
(x,t,, 0)(0) d,
(x,t,, ) x Qp 0 < t T
(x,t,, ) =Z(x , t , , ) +W(x,t,, ).
Z(x,t,y,) =
Qp
(x) exp(a0(y, )||p ) d,
|W(x,t,, )| x t
n
p
n n
u(x, t)t + (f(, )u) (x, t) = 0, x Qnp , n 1, t (0, T]u(x, 0) =(x), (f(, ))
(f(, )) (x, t) =F1x
|f()|pFx(x, t)
, x Qnp ,
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> 0 f() Qp[1, . . . , n] d
B(Qnp ) : Qnp C ||||L 0
p
u(x, t)
t + a(DTu)(x, t) =f(x, t), x Q
np , t (0, T],
u(x, 0) =(x),
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Qp
Qp n
|f()|p =
max1in
|i|pd
, Qnp ,
d f
(A) (x) =a0(x, t)(DT)(x) +
n
k=1 ak(x, t)(DkT )(x) +b(x, t)(x), >1 0< 1 < . . . < n < ak(x, t) b(x, t)
g(x) Qp[x1, . . . , xn] p |g|sp
g
|g|s
p, = Qnp\g1(0) (x)|g(x)|sp dnx, (x)> 0, S(Qnp ), g
g |g|sp
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|g|sK, ps
S(Qnp ) (ac(g))|g|sp Z
p
pm pm
Z(s,g,) =(ac(g))|g|sp,
p
E(, g) =
Qnp
(x)(g(x)) dnx;
|E(, g)|= O(||0+p ),
0 {Re(s)| s Z(s,g,)}
|f|sp
T S(Qnp ) > 0 |f|
pT = 1
E=F1TS
(Qnp ) E p
p
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p
p
p p p Qp Q p | |p
|x|p =
0 x= 0
p x Q x= pab a b p
= (x) p x ord (x) | |p p Qp
p
|x|p 0 |x|p = 0 x= 0
|xy|p= |x|p|y|p
|x + y|p (|x|p, |y|p)
Zp ={x Qp | |x|p 1} p pZp= {x Qp | |x|p< 1} Zp/pZp p Fp
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P
p x= 0
x= p
j=0xjp
j ,
= ord (x) Z xj {0, 1, . . . , p 1} x Qp {x}p
{x}p =
0, x= 0 0
p||1j=0
xjpj, 0 (x + x
) = (x) x Bnl(x)
: Qnp C C
S(Qnp ) S(Qnp ) l 0
(x + x) = (x) x Bnl x Qnp
l= l ()
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P
S(Qnp ) S(Qnp )
S(Qnp )
= (1, . . . , n) x= (x1, . . . , xn) Qnp x:=
ni=1 ixi
S(Qnp
)
(F)() =
Qnp
( x)() dnx Qnp
( x) =n
i=1(ixi) = exp
2in
i=1 {ixi}p
(xj) =
exp
2in
i=1 {xj}p
Qp
S(Qnp )
r r Z Br(0) := (prZp)
n
S
r
:= {
rN
|
|Br(0)= (0)
}.
L:= L(Qnp ) ={ S |Qnp
(x) dnx= 0} W :=W(Qnp )
C r r Z
S L + W
W = prn
Qnp
(x) dnx
r W L = W L S
L W W {F()| L}
p
f() Qp[1, . . . , n] f(D, ) >0 |f()|p
(f(D, )) =F1
|f|p F
,
S f(D, ) L2(Qnp )
(D) (x) =F1x ||p Fx
, >0, S(Qp).
p
1
2i
ddx
=
R
e2ix
Fx
() dn, N.
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P
p (n)p (s)
(n)p (s) =1 psn
1 ps s C, s= 0.
n + 2ilnpZ s= 0
(n)p (s)(n)p (n s) = 1, s / {0} {n+
2i
lnpZ}.
Rs
Rs(x) =||x||snp
(n)p (s)
, (s)> 0, s /n+2i
lnpZ, x Qnp .
C
Rs(x), (x)= 1 pn
1 psn(0) +
1 ps
1 psn
||x||p>1
||x||snp (x) dnx
+ 1 ps
1 psn ||x||p1||x||snp ((x) (0)) d
nx,
n+ 2ilnpZ (s)> 0
Rs(x), (x)= 1 ps
1 psn
Qnp
(x)||x||snp dnx, s /n+
2i
lnpZ,
Rs(x), (x)= 1 ps
1 psn
Qnp
((x) (0))||x||snp dnx.
s= 0
R0(x), (x):= lims0
Rs(x), (x)= (0),
R0(x) = (x) Rs S(Qnp ) s
C\
n+ 2ilnpZ
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P
||x||sp (s) > 0
||x||sp, (x) =
1 pn
1 psn
(0) + ||x||p>1 ||x||sp(x) d
nx
+
||x||p1
||x||sp((x) (0)) dnx, S.
||x||sp n
S(Qnp ) (F Rs) (x)
||x||sp s /n+ 2ilnpZ
DT >0
(DT)(x) =F1x
||||p Fx
S(Qnp )
(DT) (x) = (R ) (x) =
1 p
1 pn
Qnp
||y||np ((x y) (x)) dny.
(x) ||x||p1
||x||np |(x)| dnx 0 |f()|p
(f(D, )) =F1
|f|p F
,
S f(D, ) L2(Qnp )
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P
f() Qp[1, . . . , n] f() d (i) f() d (ii)f() = 0 = 0
f() Qp[1, . . . , n]
d C0(f) C1(f)
C0(f)||||dp |f()|p C1(f)||||
dp, Q
np .
f() cf() c Qp Zp
f() Qp[1, . . . , n] d A Qnp 0 /A m = m(A, f) |f()|p p
m A
A Li=1Bi Bi =zi+ (pmZp)
n |f()|p =|f(zi)|p
Bi
f() Zp[1, . . . , n] d
|f|p f(D, ) d
g(x) Qp[x] x = (x1, . . . , xn) g(x)
|g|sp, =
Qnpg1(0)
|g(x)|sp(x) dnx,
s C (s) > 0 S dnx Qnp
g(x)
|g|sp |g|sp, p
s S
EE{(NE, nE)}
EE(1 pnENEs)|g|sp S |g|sp
S |g|sp, Z(s, g) Z(s, g) =Z0(s, g)
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P
f(x) Zp[x] x= (x1, . . . , xn) d
Z(s, f) = L(ps)
1 pdsn,
L(ps) ps s= n/d Z(s, f)
A = {x Znp | (xi) d, i = 1, . . . , n} A = {x Znp | (xi) 0
|f|sp, F= Z(s, f)i
cipdris, (s)> 0.
1 pds
1 pnRds+n, p
nriri= 1 pdsn
1 pn ||x||dsp , p
nriri= pdris,
ri Z (s)> 0
|f|s
p, F=
1 pds
1 pn Z(s, f)Rds+n, F,
(s)> 0 Z(s, f) Rds+n C
C W
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u(x, t)
t + a(DTu)(x, t) =f(x, t), x Q
np , t (0, T],
u(x, 0) =(x),
a > 0 >0 DT p
Z(x, t) =
Qnp
(x )eat||||p dn.
Z(x, t) = (1 pn)||x||np k=0 qkneat(qk||x||1p ) ||x||np eat(p||x||1p ) ; Z(x, t) =
m=1
(1)m
m!1pm
1pmn(at)m||x||mnp x= 0;
Z(x, t) 0 x Qnp t (0, T].
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Z(x, t)
Z(x, t) =
k=||||p=pk(x )eat||||
p dn,
||||p=pk
(x ) dn=
pkn(1 pn), ||x||p p
k
pknpn, xp = pk+1
0, ||x||p > pk+1,
Z(x, t) = (1 pn)||x||np
k=0
pkneat(pk||x||1p ) ||x||np eat(p||x||
1p ) .
Z(x, t)
ex
Z(x, t) =
m=1
(1)m
m!
1 pm
1 pmn(at)m||x||mnp x= 0.
l(x) Bnl(0) Fl =
pnll
Z(x, t) =
l=
eatpl
||||p=pl
(x ) dn
=
l=
eatpl
(pn(l)l(x) pn(l1)l+1(x))
=
l=
pnl(eatpl
eatp(l+1)
)l(x) 0.
Z(x, t) C t(t1/ + ||x||p)n
t >0 x Qnp
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l pl1 t1/ pl
Z(x, t)
Qnp
eat||||p dn
Qnp
eap(l1)||||p dn=
Qnp
ea||p(l1)||p dn
=p(l1)n Qnp
ea||||p d= C0()pnpln C1tn/.
||x||p t1/
Z(x, t) ||x||np
m=1
Cm2m!
(t||x||p )m C3t||x||
np .
||x||p t1/
Z(x, t) C3t||x||np 2
+nC3t(t1/ + ||x||p)
n.
||x||p < t1/
Z(x, t) C1tn/ 2+nC1t(t
1/ + ||x||p)n.
Z(x, t) x L1(Q
np ) L2(Q
np )
Qnp
Z(x, t)dnx= 1.
M
M >0 (x) Qnp
|(x)| C() (1 + ||x||p).
t M
t
C
l ()
t
M <
limt0+
Qnp
Z(x , t)() dn= (x).
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Qnp
Z(x , t)() dn (x)
=
Qnp
Z(x , t) (() (x)) dn
Qnp
Z(x , t)|() (x)| dn
C
Qnp
t(t1/ + ||x ||p)n|() (x)| dn:= I(x, t) .
M < I(x, t)
I(x, t) =C
||x||p>pt(t1/ + || x||p)
n|() (x)| dn
I1(x, t) +I2(x, t),
I1(x, t) :=C1t
||x||p>p
1 + ||||p(t1/ + ||x ||p)+n
dn,
I2(x, t) :=C t|(x)|
||x||p>p
(t1/ + || x||p)n dn.
>0 t >0 I2(x, t) C2t|(x)|,
<
I1(x, t) C1t
C3+
||||p>p
||x ||p||||p+n
dn
C1t
C3+
pxp
||x ||p||||p+n
dn
=
C1tC4(x) + ||||p>xp 1||||p+n dn =C5(x) t.
limt0+
Qnp
Z(x , t)() dn (x)
limt0+ C6(x) t= 0.
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Z(x, t) 0 x Qnp t (0, T];
Qnp
Z(x, t) dnx= 1 t >0
SQnp
lim(x,t)(x0,0)
Qnp
Z(x , t) () dn= (x0)
Z(x, t+ t) =Qnp
Z(x y, t) Z(y, t) dny t t >0
eat||||p L1
Qnp
Qnp Z(x y, t1)Z(y, t2) dny= F1 (F(Z(y, t1) Z(y, t2)))
=F1
eat1||||p eat2||||
p
=Z(x, t1+ t2).
b >0 0 < x Qnp
I(b, x) =
Qnp
(b+ ||x ||p)n ||||pd
n C b
1 + ||x||p
,
C b x
m pm1 b pm
(b+ ||x ||p)n
pm1 + ||x ||p
n,
I(b, x) I(pm1, x) =
Qnp
pm1|p+ ||x ||p
n||||pd
n
=p(m1)(n)Qnp
1 + ||pm1x pm1||p
n||||pd
n
=p
(m1)() Qnp 1 + ||pm1x ||pn ||||pdn=p(m1)()I(1, pm1x).
pm1x= y ||y||p=pl
I(1, y) =I1(y) +I2(y) +I3(y),
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I1(y) =l1
k=||||p=pk
(1 + ||y ||p)n ||||pd
n,
I2(y) =||||p=pl
(1 + ||y ||p)n ||||pd
n,
I3(y) =
k=l+1
||||p=pk
(1 + ||y ||p)n ||||pd
n.
I1(y) C0(1 + ||y||p)n||y||+np ;
I2(y) C1||y||p
I2(y) C2
I(1, y) C3(1 + ||y||p)
I(b, x) C3p(m1)()(1 +p(1m)||x||p )
C3pm(1 + ||x||p) C b
1 + ||x||p
.
I1(y) =l1
k=||||p=pk(1 + ||y ||p)
n ||||pdn,
= (1 pn)(1 + ||y||p)n
l1k=
p(+n)k
C0(1 + ||y||p)n||y||+np ,
C0=(1 pn)pn
1 pn .
y Qp |
y|p= p
l =||y||p
I2(y) = ||||p=pl
(1 + ||y ||p)n
||||pdn
=||y||p
||||p=pl
1 + |y|p||y1y y1||pn dn
=||y||p
||||p=1
||y||1p + ||u ||p
ndn u=y1y.
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Am= { Qnp | ||||p = 1 ||u ||p = p
m}, m N,
I {1, 2, . . . , n}
Am,I={ Am| |ui i|p= pm i I |ui i|p < pm i /I},
u= (u1, . . . , un) = (1, . . . , n) Qnp ||||p = ||u||p= 1
Am
IAm,I
vol(Am,I) (pm(1 p1))|I|(pm1)n|I|,
|I| I
vol(Am)n
|I|=0
n
|I|
(pm(1 p1))|I|(pm1)n|I| =pmn,
I2(y) =||y||p
m=0
Am
||y||1p + ||u ||p
ndn
||y||p
m=0
||y||1p +p
mn
pmn
=||y||p1 pn
m=0
||||p=pm
||y||1p + ||||p
ndn
=||y||p1 pn
||||p1
||y||1p + ||||p
n
dn
C1||y||p
Qnp
||y||1p + ||||pn dn=C1||y||
p
Qnp
||y||1p + ||y||
1p ||y||pn dn
=C1||y||+np
Qnp
(1 + ||y||p)n dn=C1||y||
p
Qnp
(1 + ||||p)n dC1||y||
p .
I3(y) =
k=l+1
||||p=pk
(1 + ||||p)n ||||pd
n,
Qnp
(1 + ||||p)n ||||pd
n= C.
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>0
||x||p = 1
(n)p ()
Qnp
||y||np ((x y) 1) dny
x Qnp
0<
(DTZ)(x, t) =
Qnp
(x )||||peat||||p dn.
Z(x y, t) =Z(x, t) ||y||< ||x|| (DTZ)(x, t)
(DTZ)(x, t) = 1
(n)p ()
Qnp
||y||np (Z(x y, t) Z(x, t)) dny
= 1
(n)p ()
||y||p||x||p
||y||np (Z(x y, t) Z(x, t)) dny.
(DTZ)(x, t)
1
(n)p ()
||y||p||x||p
Ct||y||2np + Z(x, t)||y||
np
dny
< .
(DTZ)(x, t)
Z(m)(x, t) :=
||||ppm
(x )eat||||p dn.
Z(m)(x, t) x m (DTZ)(x, t)
(DTZ(m))(x, t) =
1
(n)p () Qnp
||y||np (Z(m)(x y, t) Z(m)(x, t)) dny
= 1
(n)p ()
||y||p>pm
||y||np (Z(m)(x y, t) Z(m)(x, t)) dny
= 1
(n)p ()
||y||p>pm
||y||np
||||ppm
eat||||p (x )((y ) 1) dn dny
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=
||||ppm
eat||||
(x )
1
(n)
p () ||y||p>pm ||y||np ((y ) 1)d
ny dn.
||y||p pm
(y ) = 1 ||||p pm
(DTZ(m))(x, t)
||||ppmeat||||
p (x )
1
(n)p ()
Qnp
||y||np ((y ) 1) dny
dn
=
||||ppm
eat||||
(x )||||pdn.
(DTZ)(x, t) =Qnp
eat||||p (x )||||pd
n.
Z
t(x, t) =a
Qnp
(x )||||p eat||||p dn;
Z
t(x, t) =a(DTZ)(x, t) 0< .
Zt (x, t) Ct1/ + ||x||pn ;(DTZ)(x, t) Ct1/ + ||x||pn .
Qnp
(DTZ)(x, t) dnx= 0.
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p
(x), f(x, t) M
0 <
u(x, t)
t + a(DTu)(x, t) =f(x, t), x Q
np , t (0, T],
u(x, 0) =(x),
a >0 >0 M
u(x, t) =
QnpZ(x , t)() dn+
t
0
Qnp
Z(x , t )f(, ) dn
d.
u1(x, t) :=
Qnp
Z(x , t)() dn,
u2(x, t) :=
t0
Qnp
Z(x , t )f(, ) dn
d .
u(x, t) M t u(x, t)
u1(x, t) M t l N (+ y) =() yp p
l
y = u1(x, t) u1(x, t) |u1(x, t)| C(1 + x)
u1(x, t) M
t
u2(x, t) x |u2(x, t)| C T(1 + x)
u(x, t) =u1(x, t) + u2(x, t) M
t
limt0+ u(x, t) =(x) limt0+ u1(x, t) =(x) lim
t0+ u
2(x, t) = 0 |u
2(x, t)| C t (1 + x) t T
u1(x, t), u2(x, t) t
u1t
(x, t) =
Qnp
Z
t(x , t)() dn.
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u2t (x, t) = t0 Qnp Zt (x , t )f(, ) f(x, ) dn d+ f(x, t).
u2,h(x, t) :=
th0
Qnp
Z(x , t )f(, ) dn
d ,
h uh(x, t+ t
) uh(x, t)
t
th0
Qnp
Z(x , t+ t ) Z(x , t )
t f(, ) dn
d
+ th+tth
Qnp
Z(x , t+ t ) Z(x , t )
t f(, ) dn
d
+1
t
th+tth
Qnp
Z(x , t )f(, ) dn
d .
t 0+ th0
Qnp
Z
t(x , t )f(, ) dn
d ,
Qnp
Z(x , t+ t ) Z(x , t )
f(, ) dn,
Qnp
Z(x , t )f(, ) dn,
Qnp
Z(x , h)f(, t h) dn.
u2,ht
(x, t) =
th0
Qnp
Z
t(x , t )f(, ) dn
d
+
Qnp
Z(x , h)f(, t h) dn.
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u2,ht
(x, t) =
th0
Qnp
Z
t(x , t )(f(, ) f(x, )) dn
d
+ th0
Qnp
Z
t(x , t )f(x, ) dn
d
+
Qnp
Z(x , h)(f(, t h) f(, t)) dn
+
Qnp
Z(x , h)f(, t) dn.
t = f
Qnp | p pm f h f(x, t) h 0+
u2t
(x, t) =
t0
Qnp
Z
t(x , t )(f(, ) f(x, t)) dn
d+ f(x, t).
u
t(x, t) =
Qnp
Z
t(x , t)() dn
+
t0
Qnp
Z
t(x , t )(f(, ) f(x, t)) dn
d+ f(x, t).
DT 0 < u(x, t) (DTu)(x, t) > u(x, t) M
<
(DTu1)(x, t) = Qnp
(DTZ)(x , t)() dn.
Z(x, t) := (DTZ)(x, t)
Z,l(x, t) := 1
(n)p ()
||y||p>pl
||y||np
Z(x y, t) Z(x, t)
dny.
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1
(n)p ()
||y||p>pl
||y||np
u1(x y, t) u1(x, t)
dny
= 1
(n)p ()
||y||p>pl
||y||np
Qnp
Z(x y , t) Z(x , t)
() dn
dny
=
Qnp
()
1
(n)p ()
||y||p>pl
||y||np
Z(x y , t) Z(x , t)
dny
dn
=
Qnp
Z,l(x , t)() dn.
m
||x||ppm
Z,l(x , t)() dn+ ||x||p m Z,l(x, t) =Z(x, t)
1
(n)p ()
||y||p>pl
||y||np
u1(x y, t) u1(x, t)
dy=
||x||ppm
Z(x , t)() dn
+ ||x||p m liml
liml
||x||p
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=
||x||p liml
<
(DTu2)(x, t) =
t0
Qnp
(DTZ)(x , t )f(, ) dn
d .
u2,h(x, t) := th
0 Qnp
Z(x y, t )f(y, ) dny d.
1
(n)p ()
||y||p>pl
||y||np
u2,h(x y, t) u2,h(x, t)
dny
= 1
(n)p ()
||y||p>pl
||y||np
th
0 Qnp Z(x y , t ) Z(x , t )
f(, ) dn
d
dny
=
th0
Qnp
Z,l(x , t )f(, ) dn
d, Z,l(x, t)
Z,l(x, t) =
Qnp
(x )Pl()eat||||p dn,
Pl() =
1
(n)p () ||y||p>pl
||y||n
p (y ) 1 dny.
|Pl()| 2||||p
|(n)p ()|
||u||p>1
||u||np dnu= C||||p ,
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|Z,l(x, t)| C.
||x ||pp(l1)
Z,l(x , t ) =Z(x , t )
th0
Qnp
Z,l(x , t )f(, ) dn
d
=
th0
||x||pp(l1)
Z(x , t )f(, ) dn
d
+
th0
||x||ppl
(DTZ)(x , t ) (f(, ) f(x, )) dn
d ,
l f(, ) u2,h M h h 0+
(DTu2)(x, t) =
t0
Qnp
(DTZ)(x , t ) (f(, ) f(x, )) dn
d.
(DTu)(x, t) =Qnp
(DTZ)(x , t)() dn
+
t0
Qnp
(DTZ)(x , t )(f(, ) f(x, )) dn
d;
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a(DTu)(x, t) =
Qnp
Z
t(x , t)() dn
t
0 QnpZ
t
(x , t )(f(, ) f(x, )) dn d , 0<
u(x, t) M t u(x, t) u(x, t)
g(x) Zp[x1, . . . , xn] g(x) Fp[x1, . . . , xn] p g(x) p f(x) Zp[x1, . . . , xn] f(0) = 0 d f(x)= 0 f(x) p
x Fnp |f(x) = 0
={0} ,
f(x) Qp x Qnp |f(x) = 0
={0} .
f p f Qp
I {1, . . . , n} fI(x) fI(x) f(x)
TI :=
x Znp |xi= 0 i I
,
TI := x Fnp |xi= 0 i I . f(x) Zp[x1, . . . , xn] f(0) = 0 d Zp f(x)
p I {1, . . . , n} fI(x) p
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f(x) =x2 y2 Zp \Zp
2
Zp
2
:=
x Zp |x = y
2 y Zp
.
f(x) p
p
n n= 1 1 n k k 2 g (x1, . . . , xk) p d Zp
l Fp l 2 f(x1, . . . , xk+1) = g (x1, . . . , xk)l xldk+1
f(x1, . . . , xk+1) p
f(x) Zp[x1, . . . , xn] f(0) = 0 d Zp f(x) p
|f(x)|p= xdp x Q
np
A:=
(z1, . . . , zn) Znp | |zi|p= 1 i
f(x)
Qp supzA
|f(z)|p
xdp |f(x)|p
infzA
|f(z)|p
xdp
|f|p|A1.
I {1, . . . , n}
AI={x A | |xi|= 1 i I} .
IAI A I {1, . . . , n}
|f|p |AI1 I
I={1, . . . , r} 1 r n x AI xi Z
p i= 1, . . . , r xi pZp i= r+1, . . . , n f(x) =fI(x)= 0
f p |f|p|AI1
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Z(x, t)
( (t) f) (x) =
Qnp
Z(x , t) f() dn.
Z(x, t)> 0 (t) f(x) 1 (t)
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p
u(x, t)
t + a0(x, t)(D
Tu)(x, t) +
nk=1 ak(x, t)(D
kT u)(x, t)
+b(x, t)u(x, t) =f(x, t), x Qnp , t (0, T],
u(x, 0) =(x).
> 1 0 < 1 < . . . < n < a0(x, t) a1(x, t), . . . , an(x, t)
b(x, t) DT
M ( 0) (x) Qnp
|(x)| C(1 + ||x||
p). M
n= 1
Z(x,t,y,)
u(x, t)
t + a0(y, )(D
Tu)(x, t) = 0, x Q
np , t (0, T],
u(x, 0) =(x),
y Qnp > 0
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P
Z(x,t,y,) =
Qnp
(x )ea0(y,)t||||p dn.
Z(x,t,y,)
Z(x,t,y,) C t(t1/+||x||p)n, Zt (x,t,y,) C(t1/+||x||p)n, (DTZ)(x,t,y,) C(t1/+||x||p)n,
y
Qnp
Z(x,t,y,) dnx= 1,
Z
t(x,t,y,) =a0(y, )
Qnp
(x )||||p ea0(y,)t||||p dn,
(DTZ)(x,t,y,) =
Qnp
(x )||||pea0(y,)t||||
p dn,
Qnp
(DTZ)(x,t,y,) dnx= 0.
ak(x, t) k= 0, 1, . . . , n b(x, t) 0 < 1 a1(x, t) = = an(x, t) = 0 0 <
n= 1
u(x, t) f(x, t) = 0 u M t u(x, 0) = 0 u(x, t) = 0 x Qnp t (0, T]
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P
u(x,t,) := t
Qnp
Z(x y, t ,y ,)f(y, ) dny d,
< t f(x, t) x Qnp (x, t) Qnp (0, T]
|f(x, t)| C t(1 + ||x||p),
0
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P
|R(x,t,, )| Cn+1
k=1(t )1/ + ||x ||p
kn,
C
J(x,,t , ) =
t
(t )/( )/Qnp
(t )1/ + ||x ||p
nb1 ( )1/ + || ||p
nb2dn
d,
0 < t x, Qnp b1, b2> 0 + b1< + b2<
J(x,,t , )
C
(t )
(++b2) B(1
, 1
+ b2
)
(t )1/ + ||x ||pnb1+
C
(t )
(++b1) B(1
+ b1
, 1
)
(t )1/ + ||x ||pnb2 ,
C b1 b2 B(z1, z2)
|Rm(x,t,, )|
C Mm(t )(m1)/((/))m
(m/)
n+1k=1
(t )1/ + ||x ||p
kn, C
m m = 1 m
|Rm+1(x,t,, )| t
Qnp |R(x,t,,)| |Rm(,,, )| dn d
=C Mm((/))m
(m/)
n+1k,l=1
t
( )(m1)/Qnp
(t )1/ + ||x ||p
kn( )1/ + || ||pln dnd.
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P
Ik,l(x,,t , ) =
t
( )(m1)/
Qnp
(t )1/ + ||x ||pkn( )1/ + || ||pln dnd.
Ik,l(x,,t , )
C B( k
,
m +
)(t )(m+k)/
(t )1/ + ||x ||p
ln+ CB(1,
m+ l
)(t )(m+l)/
(t )1/ + ||x ||pkn.
, >0
B(x + , y+ ) B(x, y)
B( k
,
m+
) B(
,m
),
B(1,m+ l
) B(
,m
),
(t )(m+k)/ C(t )(m+11)/.
|Rm+1(x,t,, )|
C Mm+1(t )m/ ((/))m+1
((m+ 1)/)
n+1k=1
(t )1/ + ||x ||p
kn.
(x,t,, ) =
m=1
Rm(x,t,, ),
|(x,t,, )| Cn+1
k=1(t )1/ + ||x ||pkn
n 0 < 1 a1(x, t), . . . , an(x, t) 0 <
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P
u(x, t)
t + a0(x, t)(D
Tu)(x, t) +
nk=1 ak(x, t)(D
kT u)(x, t)
+b(x, t)u(x, t) =f(x, t), x Qn
p , t (0, T],u(x, 0) =(x),
u(x, t) =
t0
Qnp
(x,t,, )f(, ) dn d+
Qnp
(x,t,, 0)() dn,
Qnp [0, T] t M t (x,t,, ) x, Q
np
0 < t T
(x,t,, ) =Z(x , t , , ) +W(x,t,, ),
|W(x,t,, )| C{(t )n+[(t )1/ + ||x ||p]n
+ (t )n+1k=1
[(t )1/ + ||x ||p]kn}.
u1(x, t) u2(x, t)
u1(x, t) = t0
Qnp
Z(x , t , , )f(, ) dn d
+
t0
Qnp
Z(x , t ,,)F(, ) dnd,
u2(x, t) =
Qnp
Z(x ,t,, 0)() dn
+
t
0 QnpZ(x , t ,,)G(, ) dnd,
F(, ) =
0
Qnp
(,,, )f(, ) dnd,
G(, ) =
Qnp
(,,, 0)() dn.
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P
|F(, )| C, |G(, )| C n+1/,
Qn
p
(0, T]
F
G
N ai b Zi Z || q
N
(x + ,t,+, ) =(x,t,, ),
F(+ , ) =F(, ), G(+, ) =G(, ).
u1(x, t) u2(x, t)
u(x, t)
u(x, t) (x) t 0 u2(x, t) (x) t 0
u2(x, t) =
Qnp
[Z(x ,t,, 0) Z(x ,t,x, 0)]() dn
+
Qnp
Z(x ,t,x, 0)[() (x)] dn+ (x).
Z
{| ||x ||ppN}.
t 0
(x,t,, )
(x,t,, )
b(x, t) = 0
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DTu= v, v S, >0.
E S
E v
E c E+ c
E
DT((E+ c) v) =DT((E v) + (c v))=u+ DT(c v) =u,
u c v DT
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E(x) =
1 p
1 pn||x||np =n
1 pn
pn lnpln(||x||p) = n
E FE
||x||p FE= 1,
||x||sp =mZ
cm(s+ )m
cm S
m S ||x||sp ||x||
s+p =||x||
p ||x||
sp
s=
||x||p cm= 0
m n 0< < n
||x||p1||x||np d
nx
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E, = 1 p
1 pn (Qp)n ||x||np (x) dnx
= 1 p
1 pn
Qnp
||x||np (x) dnx.
(x) = (x) S F(
)
F
F()
=
= n
c0
||x||sp, F()= mZcm, F()(s + n)
m,
||x||sp, F()= 1 ps
1 psn
Qnp
||x||snp (x) dnx
= (1 ps)
Qnp
pv(x)(s+n)
1 psn(x) dnx,
x= (x1, . . . , xn) v(x) := 1in
v(xi) ||x||p= pv(x)
(1 ps)pv(x)(s+n)
1 psn =
1 pn
lnp (s + n)
1
+(1 p1)v(x) lnp lnpp +
(p1)2p lnp
lnp + O((s + n)),
En, = c0, =
Qnp
1 pnpn ln(p)
ln(||x||p) +pn 3
2pn
(x) dnx.
F() S
L
v L >1 DTu= v =n L 1/= 1/+ /n ||u||LC||v||L
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u= Iv, (Iv) (x) = 1
()
Qnp
v(y)||x y||n dny.
f(D, ) d E f(D, )u= v >0 v S
E(x) =
L(p)(1 pd)
(1 pn)(1 pdn)||x||dnp W =n/d
L(pn/d
)(1 pn
)(1 pn)(pn lnp)
ln(||x||p) W = n/d,
L(ps) Z(s, f)
E FE
|f|p FE= 1 S.
|f|
p , =
(1 pd)L(p)
(1 pn)(1 pdn)R
d+n,
W s C W
=n/d
|F(E )(x)| C()||x||dp |F()(x)|,
x Qnp W
S l
||||2Hl =
Qnp
[ (1, ||||p)]2l|F()()|2 dn.
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S ||||Hl l Hl :=Hl(Qnp )
Hl S
f(x) = 0 ||x||p 1||x||p ||x||p > 1, > n
||f||2Hl =
||||p1
(1 pn)(1 ||||np pn)(1 pn)
p||||np
2 dn. ||f||2
Hl n/2 Hl
Hl F() L1 Qnp
|F()()| dn=
Qnp
( (1, ||||p))l
( (1, ||||p))l|F()()| dn,
Qnp
( (1, ||||p))2l dn,
l > n/2 Qnp
|F()()| dn C||||Hl .
>0 l 0 f(D, ) :Hl+d Hl
S f(D, )
||f(D, )||2Hl = Qnp [ (1, ||||p)]2l|f()|2|F()()|2 dnC1
Qnp
[ (1, ||||p)]2(l+)|F()()|2 dn= C1||||
2Hl+ .
S Hl+d
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I() :=
||||p1
||||pdn.
I() = 1 pn
1 pnn, >n.
I() =
||||p1
||||2lp|F(vW)()|2 dn
.
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vW() = prnCr() r > 0
F(vW)() =C r()
||uW||2Hl+d C(,d,n)C2p2rn ||||p1 ||||
2dp d
n
+ ||vW||2Hl+d
C(,d,n)
C1(,d,n) + ||vW||
2Hl
,
2d >n uWHl+d
F(uL) =F(vL)|f|p ,
f
|F(uL)()| C||||d|F(vL)()| .
||uL||2Hl+d
||||p1
||||2dp |F(vL)()|2 dn
+
||||p>1
||||2lp|F(vL)()|2 dn.
||vL||2Hl
0< < n/2d ||||p1
||||2dp |()|2 dn C||||L2 ,
S
||uL||2Hl+d C||F(vL)||L2+ ||vL||
2Hl .
v Hl+d f(D, )u= v v S u f(D, )u =v
f(D, )(u u) = 0, |f|p F(u u) = 0,
F(uu)() = 0 = 0 f (x)(uu)() = 0 (u u)() = 0 u u u() =u() Qnp
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f(D, )
S l
||||2Hl :=
Qnp
||||2lp|F()()|2 dn.
S | | | |Hl l Hl :=
Hl(Qnp ) Hl Hl l 0 ||||Hl ||||Hl
> 0 l 0 f(D, ) : Hl+d Hl
Ll Wl L W | | | |Hl
Hl0:= Ll + Wl Hl.
f(D, ) >0 d l f(D, ) : Hl+d Wl
v Wl {vn} Wl
v vn W
{un}
Hl+d
f(D, )un =vn
{un} Hl+d
||un um||2Hl+d C
Qnp
||||2(l+d)p ||||2dp |F(vn vm)()|
2 dn
C||vn vm||2Hl ,
u Hl+d un u f(D, ) f(D, )u= v
f(D, ) >0 d l f(D, ) : Hl+d Ll
v Ll {vn} Ll
v vn L {un} H
l+d f(D, )un = vn
{un} H
l+d
|F(un)()| C||||d|F(vn)()|,
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||un um||2Hl+d C
Qnp
||||2(l+d)p ||||2dp |F(vn vm)()|
2 dn
C||vn vm||2Hl ,
u Hl+d un u f(D, ) f(D, )u= v
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p p
p
p
p p
p
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p
p
p
p
p
p
p
p
p
p
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p
p
p