[分析] PDE

[smartlwj (下次再努力)]

                                                       3
1. Let u be a solution of the wave equation in all of R X R.

   suppose a > 0 and u(x,0) = u (x,0) = 0 for |x|>a.
                               t
   Show that there is C > 0 s.t.
                  2
            ∫   u (x,t) dx ≦ C , for all t≧0.
             R^3

  (Hint : show that there is a finite energy solution of
          w  - △w = 0  s.t. w = u.  )
           t                  t

2. 已解決
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第一題我看提示的地方， 就先假設 e(t) = (1/2)∫(w_t)^2 + |▽w|^2 dx

然後就不知道怎麼做下去了...T.T
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◆ From: 115.43.192.87
※ 編輯: smartlwj        來自: 140.116.90.149       (02/27 17:38)
推 doubleN     :If w_tt - △w = 0  and  w_t = u
→ doubleN     :let e(t) = (1/2)∫(w_t)^2 + |▽w|^2 dx
→ doubleN     :then e'(t) = ∫w_t (w_tt - △w) dx
→ doubleN     :so e'(t) = 0,  i.e.,  e(t) = C  for any t
→ doubleN     :hence ∫u^2 = ∫(w_t)^2 ≦ C