ON CONTINUOUS DEPENDENCE OF SOLUTIONS TO THE FIRST FOURIER PROBLEM WITH RESPECT TO PARAMETER

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