ಋೖ ⌦ ⇢ RN ༗քͰͳΊΒ͔ͳྖҬ ۭؒ࣍ݩ u = u ( x, t ) x 2 ⌦ , t 0 N 1 8 > < > : ut u = f(u) in ⌦ ⇥ (0, T) u = 0 on @⌦ ⇥ (0, T) u(x, 0) = u0(x) in ⌦ ͜ͷܗͷઢܗํఔࣜʹ͍ͭͯߟ͑Δɻ 5
ূ໌ 1 = sup U2H1 0 (⌦),U6⌘0 R ⌦ { |r U |2 + p' p 1 1 U 2} dx R ⌦ U 2 dx มݪཧʹΑΓ࠷େݻ༗ϨΠϦʔͱͯ͠දͤΔɻ U = '1 ͱͯ͠ࢠΛܭࢉ͢Δɻ '1 ͕ఆৗղͰ͋Δ͜ͱɺάϦʔϯͷఆཧ Λ͍ܭࢉ͢Δͱ 1 > 0 26
ূ໌ ༏ղɾྼղͷํ๏Ͱ u ! v (t ! 1), u ! 0 (t ! 1) u 2 G ͱͳΔΑ͏ͳ Λߏ͢Δɻ Ͱද͠ɺ߹͚͍ͯࣔͯ͘͠͠ɻ u := v "e (t)'1(t < 0) u := v "e 1t'1(t < 0) ߹̍ɽ ͷͱ͖ ΛదʹܾΊɺ ͱ͓͘ɻ z[v] v z[v] = 0 ͷྵΛ ", 28
ূ໌ Y. Fukao, Y. Morita, H. Ninomiya(2004) Λࢀߟʹ ͕ͨͬͯ͠ɺ༏ղɾྼղͷํ๏ʹΑΓɺ ui( x, i ) := u (| x | , i ) {ui }i2N ͱ͓͖ɺ ui ! u (i ! 1) ͕ࣔͤΔɻ ΞείϦɾΞϧπΣϥͷఆཧΛ༻͍ͯ Λߏ͢Δɻ ղͷྻ t < 0 Ͱఆٛ͞Εͨղ u ͕ଘࡏ͢Δɻ ͜͜Ͱ 30