ON A NON-LOCAL PARABOLIC PROBLEM

. The aim of this paper is to investigate the existence of solutions of a nonlocal parabolic problem. The method of upper and lower solutions and the classical maximum principle are used to obtain our results.


Introduction
In [4], Deng studied the following nonlocal boundary-value problem where is a bounded domain in K n , dQ. £ C 2 , f(x,u) is in x and C 1 in u with f(x, 0) = 0, and y) is a continuous function defined for x € dQ., yen. He first established the comparison principle for (-Pi). Then he showed the local existence of the solution and he discussed its long time behavior, assuming The results obtained in [4] generalize the result of [6].
In [10], Yin considered a problem similar to (Pi), namely, ciij,bi G C a (0 < a < 1) and L is a uniformly elliptic operator in Dt-He proved the uniqueness and the global existence of a solution of (P2) under the conditions (1.2) x) > 0 and J $(x, y)dy < p < 1. n In general the condition (1.1) is stronger than (1.2).
This result is based on the comparison principle with nonlocal conditions combined with the existence and uniqueness of a solution for the problem It has been pointed out in [10] without proof that problem (P3) has a unique solution under condition (1.2). He assumed only that h is continuous in DT-In this paper, we consider the problem (P) for all x G fi, where / : [0, +00[ x fi x M -> M is Holder continuous, $ is Holder continuous and denotes a Radon measure, a € M + and ^ is the normal derivative. We establish the global existence of solutions under the classical maximum principle and the lower and upper solutions for a linear parabolic problem. The study of the problem with nonlocal conditions is of significance. Such problems have applications in physics and other areas of applied mathematics. For example, nonlocal conditions can be applied in the theory of elasticity with better effect than the initial or Darboux conditions. The nonlocal conditions were introduced in [3] for studying of linear parabolic problems. Nonlinear differential problems of parabolic type with nonlocal conditions together with their physical interpretations were considered by Byszewski in [2]. For other results on parabolic differential equations, we refer to [5], [8] and [11]; and for parabolic systems with time delays, [9] and [11]. denotes the Holder space of exponent a G (0,1), with the norm || , u||/c+Q • In this paper, we will assume that fi is sufficiently smooth, the function is continuous in DT X M, and locally Lipschitz with respect to u. We suppose / satisfy: there exists a function 6 : DT -> M-|-which is bounded for all (t, x) G DT such that Also, we assume that the density $ is in C 1+a '-^r{DT) satisfying the compatibility condition: We first define the upper and lower solutions.

DEFINITION. A function U* G C' 2,1 (DT) n C°(DT) is called an upper solution of problem (P) if 'u?-AU*>F(t,x,U*)
in DT,
Throughout this paper, we assume that $(i, x) > 0 and JQ y)dy ^ 0 for x G dQ.. First, we give the following fundamental maximum principle.

Y. Akila
Now, we consider the following linear parabolic problem: where a > 0 and M is .a positive constant.
in Q,

2). Then the problem (L) has a unique solution u in
In addition, there exists a positive constant independent of u such that ||u||2+Q < C.

Global existence: lower and upper solutions
In [10], Yin established the existence of solution of (P2) under the condition (1.2) by using comparison principle and monotone iterative method. In this section, our purpose is to obtain the global existence of (P). We employ the method of lower and upper solutions, the maximum principle and the integral representation of the solution. The uniqueness of the solution is also proved. Then under some additional conditions, we construct the lower and upper solutions of the problem (P) and as a result, the global existence is obtained.
Let (un)n>0 be a sequence defined by: The sequence (UN) is well defined in C 2,1 (DT) H C°(DT)-In fact, the problem (3.1) is a linear problem with respect un+1. Hence using Theorem 2.2 in section 2, we will obtain for each n, a unique solution: First, we will show that U* <un < U*, for all n. It suffices to show that un < U*, for all n. The other inequality will be obtained similarly.
Prom problem (3.1) and the definition of U*, we will show that un -U* verify
By Ascoli-Arzela Theorem, we will see that (u n ) and have respectively subsequences (u n k) and (^"¿j such that u nk uin C(Dt) and <£ nfc fin C(dQ).
After passing to the limit in the integral representation, we obtain that t u{t,x) = j T(t,x; 0,0«o(x)(Z)dt + Jdri T(t,x;r) 5
Before proving the uniqueness of the solution, under some additional conditions, it is necessary to construct a lower and upper solutions of (P) by iterative method. Staring from a suitable initial iteration uq it is possible to construct a sequence {u n } successively from the modified nonlocal problem (3.1). Denote the sequences with uo = U* and uq = U* by \u n } and {u n } respectively, and refer to them as upper and lower sequences. We show that under some conditions each of the two sequences converges monotonically to a unique solution of (P). To achieve this goal we prove the following theorem:
Let w = uo -u\ = U* -u\. By (3.1), In view of Lemma 2.1, w(t,x) > 0 for all (t, x) e DT, which shows that ui < ub. A similar argument, using the property of a lower solution, gives «j > UQ. Let w^ = u\ -it follows again from Lemma 2.1, that u/ 1 ) > 0. The above conclusions show that

Mo ^ Mi < < uo-
The above conclusion of the Lemma 2.1 follows by the principle of induction. We conclude that {n"} and {un} are monotonic and uniformly bounded on DT such that U* < Mn+1 < MN < «n < UN+1 < U*, in DT-By standard argument, we claim that there exist wandu such that U,UE C 2^{ DT) n C°(DT) and lim un(t,x) = u(t, x) and lim un(t,x) -u(t,x).
To complete the proof, we need to show that u and u are respectively maximal and minimal solutions of (P), these can be easily proved by induction. Then a and b are lower and upper solution of (P) respectively. To prove the uniqueness, it is enough to show u = u on Dy. Set w = (u -u) 2 , since u<u, f(t, x, u) satisfies (2.1), we get and on IV, ,rd(u -u) a, ,, V z < 2(u -It) j i/)(«(i, y) -u(i, y))dfi(y) n (3.2) and (3.3) imply io(i,a;) < 0 in DT and from (3.4) w(t,x) = 0 on Tr, which implies u = u on Dt, the proof is thus complete.
As a consequence of Theorem 3.3, we can obtain the invariance properties of the solution of (P).
and era < ^ y)dy < ab on Ty. n Then for any UQ with a < UQ{X) < b, problem (P) has a unique solution u(t, x) such that a < u(t, x) <b in DT-Proof of Theorem 3.4. It is easy to verify that a, b are respectively the lower and upper solutions of (P). The conclusion follows from Theorem 3.3.

Long time behavior of solution
In [6], Friedman showed that if |<&(x,y)\dy < 1, for any x € fi and f(x,u) = c(x)u, (c < 0), then the solution of (Pi) decays. Moreover [10] proved that, under condition (1 .1) with f(x,u) is decreasing in u and for C > 0, a > 0, Under the assumption (1.2) and uf(t,x,u) < 0 for all (t, x) £ Dt, [10] showed that (4.1) is also true for the solution of {Pi)-In this section, we also show that (4.1) is true for any solution u(t,x) of (P), under the same condition (1.2) we employ the same method used in the proof of [