Measure-valued solutions to a nonlinear fourth-order regularization of forward-backward parabolic equations. (English) Zbl 1407.35104
The authors are interested in providing a new fourth-order regularization of forward-backward parabolic equations. a somewhat unexpected feature is that their regularization is non-linear. The work is done in one space dimension. Under general assumptions on the potentials, which include those of Perona-Malik type, the authors establish the existence of (Radon) measure-valued solutions under both natural and essential boundary conditions. It is interesting to note that if the decay at infinity of the nonlinearities is fast, then one can have local solutions whose atomic part arises and persists (in contrast to the linear fourth-order regularization) and even disappears within finite time (in contrast to pseudoparabolic regularizations).
Reviewer: Adrian Muntean (Karlstad)
MSC:
| 35K55 | Nonlinear parabolic equations |
| 35K70 | Ultraparabolic equations, pseudoparabolic equations, etc. |
| 28A33 | Spaces of measures, convergence of measures |
| 35K35 | Initial-boundary value problems for higher-order parabolic equations |
| 35M13 | Initial-boundary value problems for PDEs of mixed type |
| 35R25 | Ill-posed problems for PDEs |
Keywords:
forward-backward parabolic equations; fourth-order parabolic equations; Radon measures; Perona-Malik equationReferences:
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