Determination of parameter p(t) in Holder classes for some semilinear parabolic equations

The authors consider the problem of finding the pair (u,p) such that u_t=Au+F(x,t,u,u_x,p) in Q_T, u(x,0)=u₀(x), x in Omega , u(x,t)=u₁(x,t), on S_T= delta Omega *(0,T), and integral _{Omega 0} phi (x,t)u(x,t)dx=E(t), 0<or=t<or=T, where Q_T= Omega *(0,T), T>0, Omega is a domain in Rⁿ with smooth boundary delta Omega , Omega ₀ contained in/implied by Omega is a domain with regular boundary delta Omega ₀, u₀, u₁, phi and E and F are known functions, u_x=(u_x1,. . .,u_xn), and Au= Sigma _i,j=1ⁿ(a_ij(x,t)u_xixj with a_ij and (a_ij)_xj smooth and 0<a₀ mod xi mod ²<or=a_ij xi _i xi _i<or=a₁ mod xi mod ². Existence and uniqueness of a smooth solution pair (u,p) which depends continuously upon the data is demonstrated under certain assumptions on the data.