Dear community.

I'm looking for survey articles about recent and old results about stability for the elliptic and backward-parabolic Cauchy problem.

For example: Take $\partial_t u + a(t)\partial^2_x u = 0$ with $u(0,x)=u_0(x)$. What is the minimal regularity-assumption on $a$, in terms of a first or second order modulus of continuity to get Hölder continuous dependence of the solution on the initial data. Or: What is the best condition like $|f(t)a(t)|\leq C$ on $a\in C^1(0,T]$ to get the same result. Another question is what happens if we mix this two conditions.

Often is claimed, that Lipschitz is the optimal regularity. But I think this is, like for uniqueness, not true. At least for Zygmund there should be a similar result and also maybe for coefficients with a first order modulus of continuity $\mu$ satisfying the Osgood Condition: $\int\limits_0^1 \frac{ds}{\mu(s)}=+\infty$.

Maybe there is not so much in this direction. For hyperbolic equations there are a lot of results with this kind of conditions.

Best CJ