Henryk Ugowski
1993 Demonstratio Mathematica  
In this paper we consider the first Fourier problem with parameter A, where T is a positive constant, L is a linear parabolic operator of the second order, Eo C R n , n > 2, is a bounded domain with boundary S and Eo = Eg U S. Parameter A varies in a subset A of a Banach space (5,|-|). We discuss the continuity of solution to the problem (0.1), (0.2) with respect to A in a fixed point Ao € A (Ao being a limit point of yl). Roughly speaking, this continuity is guaranted by the continuity (in
more » ... continuity (in some integral sense) of / with respect to A in Ao and by the uniqueness of solution to problem (0.1), (0.2) for A = A 0 . Two theorems on the existence of solution to problem (0.1), (0.2) for any A G A and on its continuity with respect to A in Ao are given and proved with the aid of Green's function applying the Schauder fixed point theorem. The used methods differ essentially from that of [1], where problem (0.1), (0.2) is also considered. Moreover, our results are extensions to parabolic equations of the important result known for ordinary differential equations (see [2], Theorem 1.12 or [6], Theorem 1). Definitions and assumptions We consider problem (0.1), (0.2) with the operator L given by the formula ( 1.1) (Lu){x,t) = aij(x,t)u XiXj (x,t) + ^2bi(x,t)u Xi (x,t) + c(x, t)u(x, t) -u t (x, t). Unauthenticated Download Date | 3/5/20 1:35 PM H. Ugowski By a solution to the problem (0.1), (0.2) we always understand a regular solution, i.e. continuous in D = EQ x [0,T] and possessing in D continuous derivatives appearing in Lu. The following assumptions will be needed. (1.1) The coefficients of the operator L (with a,j = aji) are real valued functions satisfying for any EQ, i, 7 G [0, T] the conditions |Oy(x,0 -Oy(x,t)| < JVi[|*x| a + \t -t\ a ' 2 ] , \bi(x,t)-bi(x,t)\, \c(x,t)-c(x,t)\ <-/V 1 |zar| 0i , where a G (0,1), Ni > 0 are some constants and n (0.2). Unauthenticated Download Date | 3/5/20 1:35 PM Continuous dependence of solutions 309 IK-,., A) -V(;Ao)||& } < £ if A € A, | A -A01 < r/.
doi:10.1515/dema-1993-0205 fatcat:ojkmpmzvq5ddbbhqiez7hlfgam