An inverse problem for a 2D parabolic equation with nonlocal overdetermination condition

Abstract

We consider an inverse problem of identifying the time-dependent coefficient $a(t)$ in a two-dimensional parabolic equation: $$u_t=a(t)\Delta u+b_1(x,y,t)u_x+b_2(x,y,t)u_y+c(x,y,t)u+f(x,y,t),$$ $(x,y,t)\in Q_T,$ with the initial condition, Neumann boundary data and the nonlocal overdetermination condition $$\nu_1(t)u(0,y_0,t)+\nu_2(t)u(h,y_0,t)=\mu_3(t),\quad t\in[0,T],$$ where $y_0$ is a fixed number from $[0,l].$

The conditions of existence and uniqueness of the classical solution to this problem are established. For this purpose the Green function method, Schauder fixed point theorem and the theory of Volterra intergral equations are utilized.