     u t − Lu = f(x, t); (x, t) ∈ Ω × (0, T ), u| ∂Ω×(0,T ] = 0, u| t=0 = ϕ. 1.      u t − Lu = f(x, t); (x, t) ∈ Ω × (0, T ), u| ∂Ω×(0,T ] = 0, u| t=0 = ϕ. (1.1) u(x, T ; f, ϕ) = ψ T (x), x ∈ Ω (1.2) u(x, T ; f, ϕ) f ϕ n  i,j=1 (a ij (x, t)u x j + a i (x, t)u) x i + n  i=1 b i (x, t)u x i + a(x, t)u. f u u f u ψ T (x) f ∈ L 2 (Q T ) J(f) = 1 2 u(., T ; f, ϕ) − ψ T (x) 2 L 2 (Ω) (1.3) 1 J(f) + αf 2 L 2 (Q T ) (1.4) α > 0 ψ T f(x, t) = f(x) ∈ L 2 (Ω) 2. Ω R n , n ≥ 2; T > 0 Q T = Ω × (0, T ) L 2 (Ω) Ω (u, v) L 2 (Ω) =  Ω u(x)v(x)dx u L 2 (Ω) = (u, u) L 2 (Ω) ; ∀u, v ∈ L 2 (Ω) H 1,0 (Q T ) = L 2 ((0, T ); H 1 (Ω)) u(x, t) L 2 (Q T ) ∂u/∂x i , i = 1, , n Q T (u, v) H 1,0 (Q T ) =  Q T (uv + u x v x )dxdt; u H 1,0 (Q T ) = (u, u) H 1,0 (Q T ) ; ∀u, v ∈ H 1,0 (Q T ). H 1,0 0 (Q T ) = L 2 ((0, T ); H 1 0 (Q T )) H 1,0 (Q T ) ∂Q T Q T L ∞ (Q T ) Q T a ij (x, t), i, j = 1, 2, n; a i (x, t), b i (x, t), i = 1, 2, , n; a(x, t) a ij , a i , b i ∈ L ∞ (Q T ); a ij = a ji , i, j = 1, 2, , n. ν, µ νξ 2 ≤ n  i,j=1 (a ij (x, t)ξ i ξ j ≤ µξ 2 .     n  i=1 a 2 i ,     n  i=1 b 2 i , |a| ≤ µ. u(x, t) H 1 , 0 0 (Q T ) u ∈ H 1,0 0 (Q T )  Q T (−uη t + n  i,j=1 a ij u x j η x i + n  i=1 a i uη x i + n  i=1 b i u x i η + auη)dxdt =  Ω ϕη(x, 0)dx +  Q T fηdxdt, ∀η ∈ H 1,0 (Q T ). ϕ ∈ L 2 (Ω), f ∈ L 2 (Q T ) u(x, t) C([0, T ]; L 2 (Ω)) ∩ H 1,0 (Q T ). u H 1,0 (Q T ) ≤ C(ϕ L 2 (Ω) + f L 2 (Q T ) ), C Ω ψ k ( x ) H 1 0 (Ω) L 2 (Ω) u N (x, t) = N  k=1 u N k (t)ψ k (x), (2.1) (u N t , ψ l ) + n  i=1 ( n  j=1 a ij u N x j + a i u N , ψ lx i ) + n  i=1 (b i u N x i + au N , ψ l ) = (f N , ψ l ); l = 1, 2, , N. (2.2) u N l (0) = (ϕ, ψ l ); l = 1, 2, , N, (2.3) L 2 (Ω) f f N = N  k=1 (f, ψ k )ψ k , (2.4) ψ k (f N , ψ l ) = (f, ψ l ) (2.5) u N u N  L 2 (Q T ) ≤ C, C N {u N k } {u N } u N k L 2 (Q T ) u N k x , u ∈ H 1,0 0 (Q T ), k → ∞. u ∈ H 1,0 0 (Q T ) a i = b i = 0; i = 1, 2, n, J(f N ) = 1 2  N  k=1 u N k (T )ψ k − ψ δ T  2 , (2.6) ψ δ T ψ T f N ∗ J ∗ N ≤ J(f N ∗ ) ≤ J ∗ N + ε N (2.7) ε N > 0 ε N → 0 N → ∞ J ∗ N = inf f N ∈H N J(f N ) H N (ψ 1 , ψ 2 , , ψ N ). 3. f ∈ L 2 (Ω) f N = N  k=1 f k ψ k → f L 2 (Ω) N → ∞. |J(f) − J(f N )| → 0, N → ∞. J(f) − J(f N ) = u(., T ; f, ϕ) − ψ δ T  2 L 2 (Ω) − u(., T ; f N , ϕ) − ψ δ T  2 L 2 (Ω) = u(., T ; f, ϕ) − u(., T; f N , ϕ) + u(., T ; f N , ϕ) − ψ δ T  2 L 2 (Ω) − u(., T ; f N , ϕ) − ψ δ T  2 L 2 (Ω) = u(., T ; f, ϕ) − u(., T; f N , ϕ) 2 L 2 (Ω) + 2(u(., T ; f, ϕ) − u(., T ; f N , ϕ), u(., T ; f N , ϕ) − ψ δ T ) = u(., T ; f − f N , 0) 2 L 2 (Ω) + 2(u(., T ; f − f N , 0), u(., T ; f N , ϕ) − ψ δ T ) f N → f L 2 (Ω), u(., T ; f − f N , 0) 2 L 2 (Ω) → 0, u(., T ; f N , ϕ) −ψ δ T  ≤ u(., T ; f N , ϕ)  +  ψ δ T  |J(f) − J(f N )| → 0 N → ∞.  f N ∗ (u ∗ , f ∗ ) f N ∗ f ∗ L 2 (Ω) u(., t; f N ∗ , ϕ) u ∗ L 2 (Q T ). ε > 0 f ε ∈ H 1 (Ω) J ∗ ≤ J(f ε ) ≤ J ∗ + ε 2 . J ∗ = inf f∈L 2 (Ω) J(f) N ∗ N ≥ N ∗ |J(f ε ) − J(f εN )| < ε 2 . − ε 2 < J(f ε ) − J(f εN ) < ε 2 J(f εN ) < J(f ε ) + ε 2 . J ∗ N ≤ J(f εN ) < J(f ε ) + ε 2 ≤ J ∗ + ε. lim sup N→∞ J ∗ N ≤ J ∗ . J ∗ ≤ J ∗ N , lim inf N→∞ J ∗ N ≥ J ∗ . lim N→∞ J ∗ N lim N→∞ J ∗ N = J ∗ . (3.1) 0 ≤ J(f N ∗ ) − J ∗ ≤ |J(f N ∗ ) − J ∗ N | + |J ∗ N − J ∗ |. |J ∗ N − J ∗ | |J(f N ∗ ) − J ∗ N | J(f N ∗ ) → J ∗ N → ∞, f N ∗ ∈ H N J(f). J(f) f N ∗ f ∗ , H 1 0 (Ω) L 2 (Ω), f N ∗ f ∗ L 2 (Ω). u(x, t; f N ∗ , ϕ) u(x, t; f N ∗ , ϕ) L 2 (Q T ).       u t − Lu = f(x, t); (x, t) ∈ Ω × (0, T ), u| ∂Ω×(0,T ] = 0, u| t=0 = ϕ.