MAXIMUM PRINCIPLES FOR PARABOLIC SYSTEMS COUPLED IN BOTH FIRST-ORDER AND ZERO-ORDER TERMS

Some generalized maximum principles are established for linear second-order parabolic systems in which both first-order and zero-order terms are coupled.

u E C(D) N C( of the second-order elliptic system n 02U, n m Ouj rn y a,k(X) Ox,O:k+ y b,,,(x)--x,+ y c.,(x)u, O, =1,-..,rn, i,k=l i=1 j=l j=l can be bounded by a constant times the maximum of its boundary values under a "small" condition which requires that either the domain D or the coefficients b,u and c, are sufficiently small.In this paper, we have established the same kind of maximum principle for the second- Ox,Oxk Ot i,k=l i=1 j=l j=l Moreover, our parabolic version of the maximum principle holds without any "small" conditions.When the coupling occurs only in the zero-order terms (i.e., in the case of bo,j 0 for all i,j,s except when j s), the above systems are called weakly coupled systems.For weakly coupled second-order parabolic systems, similar maximum principles have been obtained by Stys [4] and Zhou [6].Under different assumptions, different maximum principles in which the components rather than the Euclidean length of the solution vector are bounded can be found in Protter and Weinberger [3] and Dow  [1].In Weinberger's paper [5], both kinds of maximum principles have been reformulated and studied in terms of invariant sets.

MAIN RESULTS.
Consider a second-order parabolic operator with real coefficients, n 0 0 M =_ a,(x, t) Ox,Ox, Ot' a, as, i,k= in a general bounded domain ft in N'x Nt (n 2 1) with the boundary 0f: 0,f U 0ll.Here 0ft is the parabolic boundary of ft and 0tf: 0ft\0,ft.We suppose that f C D x (0, T) where D is a C. ZHOU holds.system bounded domain in I" and 0 < T < (x.The operator M is assumed to be uniformly parabolic in fl; i.e., there is a constant t >0 such that tbr all (x,t) EFt and all (y,.-.,y,) in C" the inequality .,(.,t)/,z I,1 The operator M is the principal part of each equation in tlie second-order paralic Mu, + b,,j(x,t) + c,j(x,t)u 0, (2.2) = = j= We suppose that the complex-valued coefficients b,,, d all (z, t) .

c. +. +
A,b.,,b , .Klein, for some K > 0. (2.3)   r,s= l =1 k,i= Here (A,)= (A,) denotes the inverse matrix of (a,,).A solution u (u,u,...,u)is a complex-vMued C'I(u0t)C() function which satisfies (2) in ft.Here C'a() is defined the set of functions f(x,t) having all x (space) derivatives of order k d (time) derivatives of order < h continuous in ft.(2.7) This inequality holds at any point in fl U 0 where p attains a maximum.Thus p cannot achieve a positive maximum at any point in ] 0fl where the quantity in brackets in (2.7) is positive.The theorem is established.
MA. Results similar to Theorem d Corolly 2 for second-order elliptic systems were proven by Hile nd ProCter [2] (under a condition which is simil to (2.8)).Bu their maximum principle for elliptic systems only holds under the restriction tha either the domn D is sufficiently smM1 or the coefficients of the elliptic system are restricted sufficiently.Corolly 2 tells us that these restrictions can be lifted or parabolic systems.
COROLLARY 5. Let K be the same number of Theorem 4.Then, for any Ca'(a u 0,fl)cl C() solution u of the system (2.2), we have u g,a + v u g,a _< exp(KT X u 0,%n + v u or equivalently, REMARK.Under the condition that either (co),,,, is a constant matrix or (Co),,,,,, is invertible for all (x,t) , the unknowns uo, s= 1,...,m, can be eliminated from the system (2.2), (2.11), (2.12), and then a system of m(n + 1) equations in the gradient of u yields a maximum principle for c V u .
ACKNOWLEDGEMENT.The author thanks Professor G.N. Hile and the anonymous referee for some helpful suggestions and comments.

THEOM 1 .
Assume conditions (1.1) and (1.3) hold.If u is a solution of (2.2) d a is a positive C'(fl O Otfl) function, then the product a [u [2 a [u [ cannot attn a positive mimum at y point in Ot where satisfies n a-Ma 2a- a, 0 Oz > K.
We differentiate (2.2) with respect o z and t, and ge m(n + ) equations: