Abstract kinetic equations relevant to supercritical media
The abstract Hilbert space equation \((Tf)'(x)=-(Af)(x)\), \(x\geq 0\) with partial range boundary condition \((Q_+f)(0)_+\) is of considerable importance for the linear transport theory. This equation has mainly be studied under the conditions that T is injective and selfadjoint and that A is Fredholm and positive (and quite often injective). The restriction to positive A excludes many applications. In the present paper T is assumed to be bounded, injective and selfadjoint and A to be selfadjoint and Fredholm with finite negative part. The completion \(H_ A\) of the domain D(A) with respect to the form \((x,y)_ A:=(Ax,y)\) is an indefinite inner product space with finite-dimensional nonpositive part. Under the additional requirement that A be T-regular (basically, this means that the spectral resolution of \(T^{-1}A\) in the space \(H_ A\) is at no point in \({\mathbb{R}}\) unbounded) the authors derive a complete existence-and-uniqueness theory for the abstract boundary value problem. To do this, they reduce it to two subproblems: (1) a half-space-problem of the same type with strictly positive \(\tilde A\) on a finite-codimensional subspace and (2) an evolution equation on a finite-dimensional subspace Z. On the first problem many results are available; the problem (2) is solved by analysing the space Z with the indefinite inner product \([u,v]:=(Tu,v)\). The measure of nonsolvability and non-uniqueness of the problem can be expressed through the sign characteristic of \(T^{- 1}A|_ Z\) with respect to the form \([\cdot,\cdot].\) The results are applied to the one-speed neutron- and to the symmetric multigroup transport equation.
complete existence-and- uniqueness theory for the abstract boundary value problem
measure of nonsolvability and non-uniqueness
results are applied to the one-speed neutron- and to the symmetric multigroup transport equation
scientific article; zbMATH DE number 3935827
scientific article