EXISTENCE OF A SOLUTION OF A FOUR / ER NONLOCAL QUASILINEAR PARABOLIC PROBLEMa

ABST The aim of this paper is to give a theorem about the existence of a classical solution of a Fourier third nonlocal quasilinear parabolic problem. To prove this theorem, Schauder’s theorem is used. The paper is a continuation of papers [1]-[8] and the generalizations of some results from [9]-[11]. The theorem established in this paper can be applied to describe some phenomena in the theories of diffusion and heat conduction with better effects than the analogous classical theorem about the existence of a solution of the Fourier third quasilinear parabolic problem.

1 INTRO,DLIIO,,,N,   In paper [7], the author studied the uniqueness of solutions of parabolic semilinear nonlocal-boundary problems in the cylindrical domain.The coefficients of the nonlocal conditions had values belonging to the interval [-1,1] and, therefore, the problems considered were more general than the analogous parabolic initial-boundary and periodic-boundary problems.In this paper we study in the cylindrical domain, the existence of a classical solution of a Fourier third nonlocal quasilinear parabolic problem, which possesses tangent derivatives in the boundary condition.The coefficients of the nonlocal condition from this paper can belong not only to the interval [-1,1] but also to intervals containing the interval [-1,1].
Therefore, a larger class of physical phenomena can be described by the results of this paper 1Received" February, 1991.Revised: April, 1991.2Permanent address: Institute of Mathematics, Krakow University of Technology, Warszawska 31-155 Krakow, POLAND.
than by the results of paper [7].Moreover, this fundamental theorem of the paper, about the existence of the solution of the nonlocal problem, can be applied in the theories of diffusion and heat conduction with better effects than the analogous classical theorem.To prove this fundamental theorem, Schauder's theorem is used.This paper is a continuation not only of paper [7] but also of papers [1]-[6] and [8].The main result of the paper is the generalization of the Pogorzelski's result (see [11], Section 22.11), and generalizations of Chabrowski (see [9]) and Friedman (see [10], Section 7.4) results.paper.
The notation, assumptions and definitions from this section are valid throughout this Let n be any integer greater than 2. Given two points, : = (Zl,...,Zn)E R n and Y = (Yt,'", Yn) -Rn, the symbol z-Y means the Euclidian distance between z and y.The Euclidian distance between two points Pt and P2 belonging to R n is also denoted by P(Pt, P2).
To prove a theorem about the existence of a classical solution of a Fourier's third nonlocal quasilinear parabolic problem, some assumptions will be used.umptiotm: I. D: = D o x (0, T), where 0 < T < c and D o is an open and bounded domain in R n such that the boundary OD o satisfies the following Lyapunov conditions: i) For each point belonging to OD o there exists the tangent plane at this point.ii) For each points P and P2 belonging to OD o the angle (np, np2 between the normal lines rip1 and np2 to OD o at points Pt and P2 satisfies the inequality tc(rtp 1, nP2) <_ eonst.[p(P 1, P2)] hL, where h L is a constant satisfying the inequalities 0 < h L <_ 1.
iii) There exists i > 0 such that for every point P belonging to ODo, each line e parallel to the normal line to OD 0 at point P has the property that ODoglK(P,g)fqe (K(P,8) is the ball of radius di centered at point P) is equal at most to the one point.
F(z, t, Zo,..., Zn) F( , t, o,'" ", n) _< C(D)Ix-hFt-"F + CF z i_T h F (2.2) i=0 for all (z, t), , t) E D O x (0, T], z i, z E R (i O, 1,..., n), where D O is an arbitrary closed subdomain of Do; M F, M F, C F, hF, h F, IF, P are constants which do not depend on D and satisfy the inequalities MF, MF, C F > O, O < h F <_ I, O < h F <_ l, O <_ IF < 1, O <_ p < l, and C(D) is a positive constant that depends on D.
where M I is a positive constant and p is a constant from Assumption V.Moreover, the set 0 of z belonging to D o such that f(z) is the continuous function is nonempty.
Moreover, we shall need the following: umtion: X.

Do
for y E o (J = 1,2,...,k), functions F ,f ,K are given by formulae (2.20),(2.21)and (2.7), respectively, and r is the fundamental solution of the homogeneous parabolic differential equation " 02u " In the paper Z denotes the set of functions w belonging to Z such that the derivatives 0_..W_w fl__w Ozl,... Oz n are continuous in D.
3_.. DEFINITION, QF A FOUR!ER,S .,THINON, LOCAL QUASI-LINEAR PAPBOLIC PROBLEM The Fourier's third nonlocal quasilinear parabolic problem considered in the paper is formulated in the form: For the given domain D satisfying Assumptions I, II and for the given functions aij, b (i, j = 1, 2,..., n), c, F, G, g, f, K satisfying Assumptions III-X, the Fourier's third nonlocal quasilinear parabolic problem in D consists in finding a function u belonging to ZI, satisfying the differential equation F(x, t, u(x, t), Ou(x' t) Ou(z, t).
dt%-(-,... -( for (z, t) e aD O x (0, T], where for each t (0, T] the symbol a% denotes the boundary value of the transversal derivative of function u at point and for each t (0, T] the symbols dtx(i) q)   denote the boundary values of the derivatives of function u in the tangent directions tx(i) (i = 1,2,...,q) at point z, respectively.
A function u possessing the above properties is called a solution in D of the Fourier's third nonlocal quasilinear parabolic problem (3.1)-(3.3).

THEOREM. ABOUT ETENCE
In this section we prove a theorem about the existence of a solution of the Fourier's third nonlocal quasilinear parabolic problem (3.1)-(3.3)assuming that Assumptions I-X from Section 2 are satisfied.
We shall find sufficient conditions that an arbitrary point U =(Uo, Ul,...,un,) belonging to set E might be transformed by into the point aJ'U = V = (Vo, Vl,...,vn,) belonging to this set.
E is precompact.