Thanks! I got it, you said that I can only infer that $f \in C_{w}(0,T;L^{2})$! What kind of assumptions should I had to try to prove this result in the strong topology? I also have that $f\in L^{\infty}(0,T;H^{-1/2})$ . There is no way no get some interpolation result, which after I can deduce the strong continuity? Thanks in advanced!

In this case, coefficients depends on both variables, time and space. That is one a my main problems, that I didnt found any reference for such equations. However thanks for the reference. I will take a look!!

Piero thanks for your answer! Can you give me some references where I can look at this tipe of results? How can you solve the problem backwards? Is there any explicit way to compute the solution u(t,x) like for the case where g(t,x) was a constant?

I want for example that it is ill posed in the sense that the solution to the equation has Fourier coeffiecients which grow up superexponential, and therefore it cant be in any Sobolev space (like for the Backwards heat equation). How ever when I have the function $g(x,t)$ , I do not know how to compute the solution (by Fourier methods for example for the Backwards Heat equation). Thank for your comment!

For example I could suppose $g(x,t)<0$ well defined on an interval $(-a,a)$ and for a short time $t>0$. Then the problem is ill posed as for the backward heat equation? Thanks Michael.

Is just a first step for a a more complicated problem I should solve. I just dont know how to express the solution $u(x,t)$ when I got this $g(x,t)$. I know how to deal with if for example i got some constant $\kappa$ where the sign of $\kappa$ would say if the problem is ill/well posed. However I do not have a constant, but a more generic function $g(x,t)$.

Yes of course. But I have a more general problem: what kind of conditions you need to impose on g(x,t) so that the problem is ill-posed. Supose for example g(x,t) is negative on some interval (-a,a) and time (0,t'). Then is the problem ill-posed? How do you prove it ( Fourier techniques)? Thanks in advanced