Advanced Search

It is known that the nonlinear nonhomogeneous backward Cauchy problem $u_t\left(t\right)+Au\left(t\right)=f(t,u\left(t\right))$, $0\le t<\tau$ with $u\left(\tau \right)=\phi$, where $A$ is a densely defined positive self-adjoint unbounded operator on a Hilbert space, is ill-posed in the sense that small perturbations in the final value can lead to large deviations in the solution. We show, under suitable conditions on $\phi$ and $f$, that a solution of the above problem satisfies an integral equation involving the spectral representation of $A$, which is also ill-posed. Spectral truncation is used...