132 29. COMPACT 2-DIMENSIONAL ANCIENT SOLUTIONSTherefore, in terms of iJ we may rewrite a and (3 as (using also that cos 'If; = r?: 1 ,
2 2 2
sec 'If; = r 2 ~^1 , sin 'If; = i:;:~ 2 , and tan 'If; =^1 ;; , where 'If; = 'If; o a-^1 )
(29.2 12 ) (cos'lf; ·a) o a -^1 = - rvrrr + 3r-^1 VrBB - 6r-^2 voo + 3vrr - 3r-^1 vr = -ra
and
(29.213) ( cos "'' o/ · (3) o a -1 = r -2-VBBB - 3-VrrB +^9 r -1VrB - -^8 r -2-Vo = r (3-.
Finally, fr om (29.212), (29.213), and v o a-^1 = <piJ = co;;"' v, we obtain thatQoa-1 = ~ (a2+f32) oa- 1 = ~ (a 2 +,82) =Q.
4 4- Notes and commentary
For t he King-Rosenau solution or sausage model, see King [160], Rosenau
[338], and Fateev, Onofri, and Zamolodchikov [109].
Theorem 29.1 is due to Daskalopoulos, Hamilton, and Sesum [93].
See Vazquez [424] for a comprehensive treatment of the porous medium equa-
tion.
We would like to thank Frank Morgan for showing us the proof of Lemma 29. 19.