Protter--Morawetz multidimensional problems

Nedyu Popivanov ; Todor Popov ; Rudolf Scherer

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 278 (2012), pp. 188-207

Résumé

About 50 years ago M. H. Protter introduced boundary value problems that are multidimensional analogues of the classical plane Morawetz problems for equations of mixed hyperbolic-elliptic type that model transonic fluid flows. Up to now there are no general existence results for the Protter–Morawetz multidimensional problems, and an understanding of the situation is not at hand. At the same time, Protter also formulated boundary value problems in the hyperbolic part of the domain – the nonhomogeneous wave equation is studied in a $(3+1)$-D domain bounded by two characteristic cones and a non-characteristic ball. These problems could be considered as multidimensional variants of the Darboux problem in $\mathbb R^2$. In the frame of classical solvability the hyperbolic Protter problem is not Fredholm, because it has an infinite-dimensional cokernel. On the other hand, it is known that the unique generalized solution of a Protter problem may have a strong power-type singularity even for some very smooth right-hand side functions. This singularity is isolated at the vertex $O$ of the boundary light cone and does not propagate along the characteristic cone. In the general case of smooth right-hand side function, some necessary and sufficient conditions for the existence of a bounded solution are given and a priori estimates for the solution are found. The semi-Fredholm solvability of the problem is proved.