In this paper the auther considered following initialboundary value problemfor system of semilinear diffusion eguationuli-Δul=fl(ul)+gl(x

t

u

Δu)

(1.3)ul|i-0=u0l(x)

(1.4)ul|eΩ=0

l=1

2

x∈Ω Rn and corresponding initial problem by means of Galerkinmethod.The main result isSuppose that the following conditions holdi)Ω∈c2

fl∈c1

fi(s)≤c0 and for n≥4

there exists a≥1

fl satisfies|fl(s)|≤A|s|~a+B

ii) For (x

t

u

Δu)E∈Ω×R+ ×R2×R~2a

g(x

t

u

Δu) is measureable in(x

t

u

Δu) and eontinuos in(t

u

Δu)

and satisfies condition(g)and(g′)

iii) uo(x)EHto(~)NHZ(Q)

n4 or n4

a0~QnHZ(2)f] L2ao(Q)

n4

a0~~__n4

a0=min{a}. Then problem (1.8)--(1.5)has unique global strong solution u(x

t)E L'*(O

TtHto(Q)NH2(D))

u

(x

t)ELo*(O

TILz(Q))N Lz(O

TIHD).If furthermore

fl(0)=0

g satisfies conditions analogous to (g) and (gP)

then initial value problem (1.3)

(1.4) has unique global strong solution.If

finaly

let gt----uz

gt--ut

coO

thenIlull=t---.o (t--

o~) .