- MEAN VALUE INEQUALITY FOR SOLUTIONS OF HEAT EQUATIONS 311
Let
Pr~ P9 (xo, To, r, -r^2 ),
where P9 is defined in (25.7). For the cutoff function 'lj; in (25.13) we take
Tl= To -r^2 , T2 =TE [To - r^2 ,To], and D = B9 (xo,r). Let 0 < r' < r. We
further require that 'lj; satisfies(25.24) 'lj; = 1 on Pr'·
For such 'lj;, by (25.22) and supp ('lj;) C Pr, we have( f 'lj.;
(^2) v (^2) P dμ9) ( T) :::; 2CJ'f! L r f v (^2) P dμ9 dT
} B9(xo,r) }To-r^2 } B9(xo,r)
(25.25) :S; 2CJ' 0 L J Lr v^2 Pdμ9 dT
for TE [To - r^2 , To], where L is as in (25.18).
Substituting the reverse Poincare-type inequality (25.21), with Tl =To-
r2 and now T2 =To, and (25.25) into (25.23), we obtain
(25.26)
1
TO 1 2(n+2)
i'lj;vPln dμ9dT
To-r^2 B9(xo,r)
< - e^0 s(l+VKr) VoC~ g B-g (xo ' r)
2
x (2CJ' 0 L j" { v^2 P dμ 9 dT) :;::;; ro { ( r^2 60 +1 L + 'lj.;^2 ) v^2 P dμ9dT.
} Pr }To-r^2 } B9(xo,r)Next, by (25.26) while using 'lj; = 1 in Pr' and 'lj.;^2 :S 1 in Pr, we have
J 1 V n!22Pdμ9 dT
pr,:S; J Lr i'lj;vPI 2(n:2) dμ9 dT
:::; e^0 s(i+VR'r) Vol§~ B 9 (xo, r) (r^260 +i L + 1)
2
x (2C'Q L j" f v^2 P dμ9 dT) :;::;; ra f v^2 P dμ9 dT
}pr }To-r2 JB9(xo,r)
n+2
= M (!Lr v^2 Pdμ9dT) 1',where
2
(25.27) M ~ e^0 s(l+VR'r) Vol§~ B9 (xo, r) (r^260 +l L + 1) ( 200 L):;::;;,
where Re (g) ;:::: -Kinn, Co is as in (25.5), c is given by Proposition 25.4,