| Abstract: |
| Forward-backward parabolic equations are well-known to be ill posed. Various types of regularisation have therefore been proposed, such as a pseudo-parabolic (PP) and a linear fourth-order (L-IV) one. Due to the intrinsically unstable character of backward parabolic equations, one may reasonably expect that the qualitative behaviour of approximating solutions strongly depends on the type of regularisation. In one space dimension, I will introduce a {\em nonlinear} fourth-order regularization and I will present some first results of a joint program with M. Bertsch and A. Tesei. In particular, I will show that this regularization differs from the above-mentioned ones in two aspects: one one hand, unlike (L-IV), it admits solutions with singularities; on the other hand, unlike (PP), singularities can both appear {\em and} disappear. We hope that these features will permit a better description of some qualitative properties of the limiting solutions, e.g. in terms of time scales. |
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