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On biharmonic evolution equations


In this project we want to consider biharmonic evolution equations on domains with Dirichlet boundary conditions. This evolution problem gives a model for the continuum theory of some special kinds of liquid crystals. The big difference between this biharmonic evolution problem and the corresponding second order problem is that in the higher order case the maximum principle does not hold. A solution of the biharmonic evolution equations in the whole space with a non-negative initial condition may change sign instantaneously. Our interest in studying the biharmonic evolution problem is to find out to what extent results proved for the heat equation really depend upon the maximum principle or in other words, upon positivity preservation. In particular w e would like to get estimates of the solutions both from above and from below. In order to find such bounds one method is to find optimal estimates both from above and from below of the semi group kernel associated. A study of the behaviour, or more specific ally of the sign of the semigroup kernel, will be a powerful tool to get bounds of the kernel itself. Optimal a priori estimates of the semigroup kernel associated to the biharmonic evolution problem is an interesting subject on its own. Moreover, these bounds turn out to be important results for the study of higher order non-linear evolution equations and also for proving boundedness and compactness of operators.

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