In this thesis, we study the global existence and the global nonexistence (blow up in finite time) of nonnegative solutions of the initial value problem for a weakly coupled system of reaction-diffusion equations ≤ft\\eqalignu[subscript]t &= L[subscript]1u + t[superscript]s[subscript]1v[superscript]p[subscript]1 v[subscript]t &= L[subscript]2v + t[superscript]s[subscript]2u[superscript]p[subscript]2 . ( x,t)ϵ IR[superscript]N x (0,T);\eqalignu( x,0)&= u[subscript]0( x) ≥ 0 v( x,0)&=v[subscript]0( x)≥ 0 xϵ IR[superscript]Nwhere s[subscript]1,s[subscript]2≥0, p[subscript]1,p[subscript]2≥1 with p[subscript]1p[subscript]2 > 1, T ≤ [infinity] (the length of the existence interval) and the L[subscript]1,L[subscript]2 are uniformly elliptic second order differential operators with uniformly bounded coefficients. We present several results for the systems:;( S[subscript][delta])≤ft\\eqalignu[subscript]t &= [delta][delta] u + v[superscript]p[subscript]1 v[subscript]t &= [delta] v + u[superscript]p[subscript]2 . ( x, t)ϵ IR[superscript]N x (0,T)where 0 ≤ [delta] ≤ 1 and p[subscript]1,p[subscript]2 ≥ 1 with p[subscript]1p[subscript]2 > 1; ( S[subscript]L)≤ft\\eqalignu[subscript]t &= L[subscript]1u + v[superscript]p[subscript]1 v[subscript]t &= L[subscript]2v + u[superscript]p[subscript]2 . ( x, t)ϵ IR[superscript]N x (0,T)where p[subscript]1,p[subscript]2 ≥ 1 with p[subscript]1p[subscript]2 > 1 and ≤ft\\eqalignL[subscript]1u &≡ [sigma][subscript]spi,j=1N [partial][over] [partial] x[subscript] i≤ft(a[subscript]ij( x)[partial] u[over][partial] x[subscript] j) L[subscript]2v &≡ [sigma][subscript]spi,j=1N [partial][over] [partial] x[subscript] i≤ft(b[subscript]ij( x)[partial] v[over][partial] x[subscript] j) . x ϵ IR[superscript]Nwith the following assumptions:;(i) the coefficients a[subscript]ij, b[subscript]ij are sufficiently smooth (a[subscript]ij,b[subscript]ijϵ C[superscript][infinity](IR[superscript]N)). (ii) a[subscript]ij = a[subscript]ji; b[subscript]ij = b[subscript]ji. (iii) there exists a constant [nu]≥1 such that ≤ft\\eqalign[nu][superscript]-1 ǁ [xi] ǁ [superscript]2 &≤ [sigma][subscript]spi,j=1N a[subscript]ij( x)[xi][subscript]i[xi][subscript]j ≤ [nu] ǁ [xi] ǁ [superscript]2 [nu][superscript]-1 ǁ [xi] ǁ [superscript]2 &≤ [sigma][subscript]spi,j=1N b[subscript]ij( x)[xi][subscript]i[xi][subscript]j ≤ [nu] ǁ [xi] ǁ [superscript]2 . x, [xi] ϵ IR[superscript]N.;( S[subscript]t)≤ft\\eqalignu[subscript]t &= [delta] u + t[superscript]s[subscript]1v[superscript]p[subscript]1 v[subscript]t &= [delta] v + t[superscript]s[subscript]2u[superscript]p[subscript]2 . ( x,t)ϵ IR[superscript]N x (0,T)where s[subscript]1,s[subscript]2≥0 and p[subscript]1,p[subscript]2≥1 with p[subscript]1p[subscript]2>1.;( S[subscript]Lt) ≤ft\\eqalignu[subscript]t &= L[subscript]1u + t[superscript]s[subscript]1v[superscript]p[subscript]1 v[subscript]t &= L[subscript]2v + t[superscript]s[subscript]2u[superscript]p[subscript]2 . ( x,t)ϵ IR[superscript]N x (0,T)where s[subscript]1,s[subscript]2≥0 and p[subscript]1,p[subscript]2≥1 with p[subscript]1p[subscript]2>1. ( S[subscript]k\ell) ≤ft\\eqalignu[subscript]t &= u[subscript]x[subscript]i[subscript]1x[subscript]i[subscript]1 +·s+ u[subscript]x[subscript]i[subscript] kx[subscript]i[subscript] k + v[superscript]p[subscript]1 v[subscript]t &= v[subscript]x[subscript]j[subscript]1x[subscript]j[subscript]1 +·s+ v[subscript]x[subscript]j[subscript] \ellx[subscript]j[subscript] \ell+ u[superscript]p[subscript]2 . ( x,t)ϵ IR[superscript]N x (0,T)where 1≤ k, \ell ≤ N, \min(k,\ell) 1.