IMPULSIVE NONLOCAL NONLINEAR PARABOLIC DIFFERENTIAL PROBLEMS

The aim of the paper is to prove a theorem about a weak impulsive nonlinear parabolic differential inequality together with weak impulsive nonlocal nonlinear inequalities. A weak maximum principle for an impulsive nonlinear parabolic differential inequality together with weak impulsive nonlocal nonlinear inequalities and an uniqueness criterion for the existence of the classical solution of an impulsive nonlocal nonlinear parabolic differential problem are obtained as a consequence of the theorem about the weak impulsive nonlinear parabolic differential inequality together with weak impulsive nonlocal nonlinear inequalities.


I. INTRODUCTION
In this paper we prove a theorem about a weak impulsive nonlinear para- bolic differential inequality together with weak impulsive nonlocal nonlinear in- equalities.The impulsive inequality, studied here, is of the form u f(t, x, u, u, u) < v f(t, x, v, v, v), s-1 where (t,x) e(uDn [(h,h+,)xR"])u (Dn[(,,to+T]x,"]), is fixed natural number, D xs one of two relatively arbitrary sets more general than the cylindrical domain (to, t o + T] x Do C N" + and t o < t x < t 2 <... < t s < t o + T.
As a consequence of the theorem about the weak impulsive nonlinear parabolic differential inequality together with weak impulsive nonlocal nonlinear inequalities we obtain a weak maximum principle for an impulsive nonlinear parabolic differential inequality together with weak impulsive nonlocal nonlinear inequalities and an uniqueness criterion for the existence of the classical solution of an impulsive nonlocal nonlinear parabolic differential problem.
Many processes in the theories of heat conduction and diffusion are characterized by the fact that at certain moments tl, t:,...,t of time they experience changes of temperatures of a heated substance or changes of amounts of a diffused substance.Moreover, for many above processes we know the relations between the temperatures of the heated substance and we know the relations between the amounts of the diffused substance at the points ti, Ti,2j_ , T,2j (j e I, i-O, 1,...,s-1) and t,,T,,2_,T,,2 (j e I;).Consequently, it is natural to assume that these changes act in the form of impulses at the points t, t2,..., t, and that the following impulsive nonlocal conditions are considered + e ., = 0, where (i-0,1,.. s) are given real functions defined on St .., respectively.
It is easy to see from (1.1) that these conditions are more general than the standard initial conditions.Moreover, if G,, j(x, u): u(T,,:i, x) for x e St, or Ti,2j Ti,2j-1 u(r, x)dr for x then conditions (1.1) are reduced to the impulsive periodic conditions and to the impulsive antiperiodic conditions, or to the impulsive average periodic conditions and to the impulsive average antiperiodic conditions for suitable functions hi, To obtain physical interpretations of the impulsive nonlocal problems considered in the paper i is enough to join he physical interpretations of the nonlocal problems and of the impulsive problems.For this purpose, compare papers [2] and [3], where physical interpretations of the nonlocal problems and of the impulsive problems were given separately.
The paper is a continuation and a generalization of papers [11-[31.Moreover, the paper generalizes some theorems from [4] and [5].To prove the main result of this paper a strong maximum principle from the author publication [1] is used.

PRELIMINARIES
The notation, definitions and assumptions given in this section are valid throughout the paper.
Let t o be real finite number, 0 < T < ec and let x-(x,...,x,) N".A bounded or unbounded set D contained in (to, t o+ T]xN" and satisfying the conditions" () (to, to + T). () The projection of the interior of D on the t-axis is the interval For any , ) D there exists a positive number r such that is said to {(t, ): (t 7 ) + (, u,) < , i=1 be a set of type (P).t<7}CD, For any t [to, t o + T] we define the following sets: It is easy to see, by condition (b) of the definition of a set of type (P), that St and o,t, where [o, o + T], are open sets in N' and N" + 1, respectively.
By s we denote a fixed number belonging o N or N o.
Let tx, t,..., t, (s N) be given real numbers such that We introduce he following ses" For an arbitrary fixed point (,)e D we denote by S-(,) the set of points (t, x)E D that can be joined with (, ) by a polygonal line contained in D along which the t-coordinate is weakly increasing from (t,x) to ( , ).
By PC(D) we denote the space of functions w: D 9 (t, z)w(t, x) e such that w is continuous in D\a(s) (s e N0), the finite limits w(t[-,x), w(ti+,x) (i = 1,..., s) exist for all admissible x e N" if s e N and w(ti, z): = w(ti + ,x) (i = 1,..., s) for all admissible z N" if s N.
We say that w PC'(D) if w PC(D)and wt, w,w = [wi],, are e The symbol M,x,,(N) is used for the space of real square symmetric matrices r = [rik], x ,.
Functions u and v belonging to PC,(D) are called solutions of the differential inequality (Pu)(t, x) <_ (Pv)(t, x), (t, x) e D(s) in D(s) (s e [No), if they stisfy (2.1) for 11 (t, x) e D(s) (s e No).
The function f is said to be uniformly parabolic in a subset E C D(s) (s e No) with respect to a function.wPC'(D)if there exists a contact > 0 (depending on E) such that for any two matrices and for (t,x) E we have <_ Ff(t, x, w(t, x), w(t, x), F) f(t, x, w(t, x), w(t, x), ) >_ x (?ii Yii), (2.2) i=l where <_ ?means that ('Yjk-rk)AjA <--0 for every (11,...,A,) N'. j,k=l If (2.2) is satisfied with x 0 for = w(t, z) and F = w(t, z)+ r, where r > 0, then f is called parabolic with respect to w in E.
Let us define the sets: =.
A bounded set D of type (P) satisfying condition (a)of the definition of a set of type (Pzsr) is called a set of type (PtsB)" It is easy to see that if D is a set of type (PIsB), 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 [D o is an unbounded essential subset] of [", then D = (o, to + T] x Do is a set of type (Pls) [(Psr), respectively].
3. A THEOREM ABOUT A WEAK INEQUALITY Now, we shall prove Theorem 3.1 which is the main result of this paper.
D is a set of type (Plsr). 2.
The function f is weakly decreasing with respect to z.
there exists a positive constant L such that the inequality Moreover, f(t,x,z,q,r) f(t,x, , ,) is satisfied for all (t, x)   3.
Since, by (a)(il) of the definition of a set of type (Ptsr), To, > to, we get, from (3.11) that condition (3.16) contradicts condition (3.6).This completes the proof of inequality (3.3) if I; is a finite set.