Local Operator Methods and Time Dependent Parabolic Equations on Non-cylindrical Domains

We investigate time dependent parabolic problems of diiusion type on open subsets of R N+1 and on networks, where the domains are possibly unbounded or non-cylindrical. The coeecients are assumed to be continuous and may be singular or degenerate at the boundary. We are looking for solutions which belong locally to suitable Sobolev spaces and vanish at the boundary. The well-posedness of the homogeneous linear problem is characterized by a barrier condition which is veriied for a large class of highly singular domains. Using this result, we solve the inhomogeneous linear equation and obtain global solutions for Lipschitz nonlinearities of, e.g., logistic type. These applications are based on an abstract approach in the framework of local operators. In this context we derive maximum principles and characterize the well-posedness of the Cauchy problem by excessive barriers and Cauchy barriers. In the parabolic case, we construct a `variable space propagator' using an associated`space-time semigroup'. The propagator then allows to solve the above mentioned problems.