Inverse problems with final overdetermination for parabolic equations with unknown coefficients multiplying the highest derivative (Q656330)

The author considers the following inverse problem \[ c(x,t,u)u_t- (a(x,t,u)u_x)_x +b(x,t,u)u_x+d(x,t,u)=f(x,t), \quad 0\leq x\leq l, \, 0\leq t\leq T \] \[ u(0,t)=v_0(t),\quad u(l,t)=v_1(t),\quad u(x,0)=\varphi(x). \] Existence and uniqueness of the solution \(u(x,t)\) satisfying the additional condition \(u(x,T)=g(x)\) is studied, assuming that the function \(a(x,t,u)\) has one of the following forms: \[ a= k(x)a_1(x,t,u),\;a=k(u)a_1(x,t,u),\;a=k(x,u)a_1(x,t,u) \] with given functions \(g(x) \) and \(a_1(x,t,u).\)