Determination of a parameter p(t) in some quasi-linear parabolic differential equations

The authors consider the following inverse problem of finding the evolution parameter p(t) and the solution u(x, t) such that ut= Sigmai,j=1n(aij(x, t)uxi+bj(x, t, u))xj+F(x, t, u, p) in QTu(x, 0)=u0(x) x in Omega Sigmai,j=1n(aij(x, t)uxi+bj(x, t, u))*vj(x)=g(x, t, u) on STand integralOmegaphi (x, t)u(x, t)dx=E(t) 0<or=t<or=T where QT= Omega *(0, T), T>0 and Omega is an open bounded region in Rnwith boundary delta Omega as smooth as needed throughout this paper; v(x)=(v1(x), v2(x),. . ., vn(x)) is the outwardly pointing normal direction on delta Omega ; u0, g, F, aij, bj, phi and E are given functions. The notion of a weak solution for the pair (u, p) is formulated. The existence, uniqueness and continuous dependence upon the data of the solution (u, p) are demonstrated for F(x, t, u)=G(x, t, u)+H(x, t)p(t) and F(x, t, u)=G(x, t, u)+u(x, t)p(t).