Determination of a parameter p(t) in some quasi-linear parabolic differential equations
John Rozier Cannon, Yuan Sheng Lin · Inverse Problems · 1988
The authors consider the following inverse problem of finding the evolution parameter p(t) and the solution u(x, t) such that u t = Sigma i,j=1 n (a ij (x, t)u xi +b j (x, t, u)) xj +F(x, t, u, p) in Q T u(x, 0)=u 0 (x) x in Omega Sigma i,j=1 n (a ij (x, t)u xi +b j (x, t, u))*v j (x)=g(x, t, u) on S T and integral Omega phi (x, t)u(x, t)dx=E(t) 0 0 and Omega is an open bounded region in R n with boundary delta Omega as smooth as needed throughout this paper; v(x)=(v 1 (x), v 2 (x),. . ., v n (x)) is the outwardly pointing normal direction on delta Omega ; u 0 , g, F, a ij , b j , 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).