Linear and Semilinear Partial Differential Equations

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Abstract Tataru 9622942 The proposed research focuses on three interconnecting subjects, namely (i) unique continuation for solutions to nonelliptic partial differential equations, (ii) semilinear (mostly hyperbolic) partial differential equations and (iii) hyperbolic mixed initial-boundary value problems. The research in unique continuation is oriented towards semilinear problems and problems with unbounded potentials. In the area of semilinear hyperbolic equations, the main effort will be directed towards the study of a class of weighted Sobolev spaces attached to (mostly) hyperbolic partial differential operators. These spaces adapted to the hyperbolic operator seem to be the natural setting for the study of semilinear equations; it appears that they are the key to further progress in the field. A main goal is to understand the multiplicative structure of these spaces, including null forms estimates. The first aim of the research on hyperbolic boundary value problems is to complete the ongoing PI's study of the trace regularity properties for solutions to such problems. This is a key step for the second part, whose goal is to understand the Strichartz type estimates for solutions to hyperbolic boundary value problems, and subsequently, to use this in the study of semilinear hyperbolic boundary value problems. It is the PI's hope that in the long run this research program could also lead to progress in the study of quasilinear hyperbolic equations. A partial differential equation is an equation which describes the relation between rates of change of quantities which depend on more than one variable. Such equations are used to model many phenomenas in physics, economics, etc. The aim of this project is to investigate some specific properties of solutions to certain partial differential equations. Part of the work is motivated by problems arising in control theory and in inverse problems; there one tries to reconstruct information about either the state of a system or th e equations governing the system, based on partial observations. Another goal of the project is to reach a better understanding of some fundamental equations which insofar have proved extremely difficult to study; for instance, Einstein's equations, which describe the structure of the relativistic space-time.