I have tried to apply mathematical constructs like parametrices and weak solutions from studies of differential equations to quantum fields in curved spacetime. Particular examples of such spacetime were studied in the thesis work. They involved the astrophysical scenario of unending gravitational collapse. Quantum fields on those spacetimes indicate the presence of very high energy all over the Cauchy horizon if it gets formed. I have also tried to study non-analytic behaviour of geodesics in the same spacetime examples at uncovered spacetime singularities. This involved mainly hyperbolic equations and non-linear geodesic equations.