On the optimal regularity of a coupled partial differential equation in a time dependent domain with conic points

We consider the heat equation in a time dependent domain with conic points coupled with a generalized string model that prescribes the motion of the boundary of the domain. This model problem is motivated by a fluid-structure interaction model. We prove the existence of a weak solution using the method of Rothe combined with a fixed point argument. Moreover, we determine optimal regularityin weighted Sobolev spaces near the conic points using integral inequalities of Hardy type and specific calculations caused by the generalized string model. The interest ofour results varies in certain subjects. On the one hand we give a general approach to deal with coupled differential equations and problems in domains with time dependent apertures. On the other hand it will be interesting to transfer both results, existence and regulartiy, to fluid structure interaction models