STRONG MAXIMUM PRINCIPLES FOR PARABOLIC NONLINEAR PROBLEMS WITH NONLOCAL INEQUALITIES TOGETHER WITH INTEGRALS

In [4] and [5], the author studied strong maximum principles for nonlinear parabolic problems with initial and nonlocal inequalities, respectively. Our purpose here is to extend results in [4] and [5] to strong maximum principles for nonlinear parabolic problems with nonlocal inequalities together with integrals. The results obtained in this paper can be applied in the theories of diffusion and heat conduction, since considered here integrals in nonlocal inequalities can be interpreted as mean amounts of the diffused substance or mean temperatures of the investigated medium.


INTRODUCTION
In this paper we give a theorem on strong maximum principles for problems with a diagonal system of nonlinear parabolic functional-differential inequalities and with nonlocal inequalities together with integrals.The diagonal system of the inequalities considered here is of the following form" (1.1) u(t,x) <_ f (t,x,u(t,x) ,Ux(t,x),Uxx (tx),u) (i = 1 ,...,m), where (t,x) D c (to,t o + T] x R n and D is one of six relatively arbitrary sets more general than the cylindrical domain (to,t o + 7] x D O c R '/.The symbol u denotes the mapping U" J (t,X) U(t,X) = (ul(t,X),...,um(t,X)) .Rm, where 13 is an arbitrary set contained in (-oo, t o + T] x R n such that c 13.The fight-hand (t,x) = gradxui(t,x) sides fi (i = 1,...,m) of system (1.1) are functionals of u; u x (i = 1,...,m) and Uxx(t,x ) (i = 1,...,m) denote the matrices of second order derivatives with respect to x of ui(tx) (i = 1,...,m).
The nonlocal inequalities together with integrals, considered here, are of the form:
The results obtained in this paper are a continuation and direct generalizations of those given by the author [5] and [4].Moreover, some results obtained here are direct generalizations of results given by Chabrowski [7].Finally, some results obtained in the paper are indirect generalizations of those given by Chabrowski [6], Walter [15] and [ 16], Besala [2], Szarski [ 14], and Redheffer and Walter [13].The method of the proof of the main theorem in this paper is similar to the method used in [5], and for ease in comparison of these methods we use in this article similar notation as in [5].If the nonlocal inequalities considered here are initial inequalities, then the results obtained in this paper are reduced to those from [4] and are based on the publication of the author [3].
Let t o be a real finite number and let 0 < T < ,,,,.A set D c {(t,x)" t >t 0, x Rn} (bounded or unbounded) is called a set of type (P) if: For any t [to, t0+ T] we define the following sets" and int {XRn:R': o.
)C: 1.We introduce the For an arbitrary fixed point (',)e D we denote by S'(7",) the set of points (t,x)e D that can be joined with (7',.)by a polygonal line contained in D along which the t-coordinate is weakly increasing from (t,x) to (7,).
In the set of mappings bounded from above in ][3 and belonging to Z,,,(13) we define the functional [wit_ max sup{0,wi(',x): (',x) 13, 7< t} where t < t o + T. i= 1...m By X we denote a fixed subset (not necessarily a linear subspace) of Zm()) and by Mnxn(R) we denote the space of real square symmetric matrices r = [rj]nxn.
For any set Z c I3 and for a mapping u X we use the symbol max u(t,x) in the sense" (t,x)z ( max ul(t,x), max um(t,x)) Let us define the following set: where I is a countable set of all such mutually different natural numbers that: (i) to < Z2i.1 < Z2i l 0 + T for i e i and T:zi.T2j. , T:z , T2j for ij e I, i j, To: = inf{T2i.1" i e I} > to if card I = R0, (iii) St St o for every t /eL)/[T2i.1 ,T2i ], (iv) St St o for every t e [T0,t 0 + T] if card I = 80- An unbounded set D of type (P) is called a set of type (Psr) (see Fig. 1) if: (a) ,5 # , (b) Fc3 Let ,5, denote a non-empty subset of,5.We define the following set: A bounded set D of type (P) satisfying condition (a) of the definition of a set of type (Psr) is called a set of type (Psi).It is easy to see that if D is a set of type (Psi), then D satisfies condition (b) of the definition of a set of type (Psr).Moreover, it is obvious that if D O is a bounded subset [Do is an unbounded essential subset] of Rn, then D = (t 0, to+ T] x Do is a set of type (Psi) [(Psr), respectively].
3. STRONG MAXIMUM PRINCIPLES WITH NONLOCAL INEQUALITIES TOGETHER WITH INTEGRALS IN SETS OF TYPES (Psi') AND (PSI3).
Our main result is the following theorem on strong maximum principles with nonlocal inequalities together with integrals in sets of types (Psr) and (Psa): Theorem 3.1" Assume that: (1) (2) D is a set of type (Psr) or (Psa).
Moreover, if there is a point ('{) D such that u(7) = max_ u(t,x), then (t,x)e D u(t,x)= max u(t,x) for (tc)e S'('{2) (t,x)e l" Proof'.We shall prove Theorem 3.1 for a set of type (Psr) only since the proof of this theorem for a set of type (Psi) is analogous.
Since each set of type (Psr') is a set of type (Pzr) from [5] then, in the case if Y hi(x) = 0 for x Sto, Theorem 3.1 from this paper is a consequence of Theorem 3.1 of ii, [5].Therefore, we shall prove Theorem 3.1 only in the case if the following condition holds: (3.5) -1 < hi(x) < 0 for x Sto. iI.
for xe Sto(m=l), x e Sto (i e I,, re=l)   are the mean amounts of this gas on the intervals [T2i.,T2i] (i I,), respectively.
Therefore, Theorem 3.1 seems to be more useful in some physical applications than Theorem 4.1 from [4] on strong maximum principles with initial inequalities of the form: U(to,X) <K for xe S o Let us observe that Theorem 3.1 from the paper is also more useful in some physical applications than Theorem 3.1 from [5], since considered here inequalities (3.2) are more sensitive to measurements than the following inequalities: [uJ( t O, x*)-KJ] + hi(x*) [laJ(T i, x) KJ] < 0 for x e Sto (j = 1,...,m)   ieI, given by the author in [5].
If I, = { 1 }, T = t o + T -At, 0 < zt < T, T 2 = t o + T, -1 < hi(x) = -h(x) < 0 for x e Sto and m=l, then the nonlocal conditions: Z2/ uJ( t'x) + ' i e I , T2li(x 2i-T2"iluJ(Tr'x)dT =0 for x Sto (/= 1,...,m) are reduced to the following condition: (5.1) to+T u(tO,x) = h(x) fu(,x)d' for x e Sto (m = 1) At to+T.At and this condition can be used to the description of heat effects in atomic reactors.It is easy to see, by (5.1), that if u(t o, x) is interpreted as the given temperature in an atomic reactor at the initial instant t 0, then the atomic reaction is the safest for 1 = h(x) < 1 and this reaction is the most dangerous for 0 < h(x) = O.In the case if h(x) = 1 for x S to, formula (5.1) is reduced to the condition:

Figure 1 .
Figure 1.The set D of type (Psr) if D = (int D) w tSto / r, I = 1,2,3,4 and to<T <T2<T3<T4 = o + T x) } for t= t 0, It is easy to see, by condition 2 of the definition of a set of type (P), that S and 8 are open sets in R n and R/, respectively.
Let 13 be a set contained in (_oo, t o + T] x R n such that following sets: =ci r: