A non-local boundary value problem method for parabolic equations backward in time (Q932354)

Let \(H\) be a Hilbert space, \(A\) be a positive self-adjoint unbounded operator. Let \(\varepsilon< E\) and \(T\) three given positive numbers. One considers the problem of finding a function \(u: [0,T]\to H\) such that for \(f\in H\), \[ u_t+ Au= 0,\quad 0< t< T,\quad\| u(T)- f\|\leq\varepsilon, \] subject to the constraint \[ \| u(0)\|\leq E.\tag{\(*\)} \] The authors regularize this ill-posed problem by the non-local boundary value problem \[ u_t+ Au= 0,\quad 0< t< T,\quad\alpha u(0)+ u(T)= f,\quad\alpha> 0, \] and obtain some error estimates. If instead of \((*)\) one has a stronger condition, involving the eigenvalues and the eigenvectors of \(A\), they get an error estimate of Hölder type in \((0,T)\) and logarithmic type at \(t=0\), or even of Hölder type in \([0,T)\).