Solvability of the initial-boundary value problem for an integrodifferential equation

Abstract

Under study is the well-posedness of the Cauchy problem for the nonstationary radiation transfer equation with generalized matching conditions at the interface between the media. We prove the existence of a unique strongly continuous semigroup of resolvents, estimate its order of growth, and consider the question of stabilization of the nonstationary solution.

Nonstationary radiation transfer through a multilayered medium with reflection and refraction conditions

“…Proof. Let I ∈  1 ( ,T ) be a solution to problem (32), (33). Then, for almost all ( , x ′ , t ′ ) ∈ S − ,T the function I ,x ′ ,t ′ belongs to the space AC[0, + ,T ( , x ′ , t ′ )] and is a solution to the problem…”

“…Let a function I be represented by formula (34). Then, for almost all ( , ( ,x,t) I( , x, t) satisfies the equalitỹ (32), (33), with data…”

“…Similar problems for the nonstationary radiation transfer equation are studied in much less detail() (1D case),() (3D case).…”

“…It this paper, we consider the problem describing a nonstationary radiation transfer in multilayered medium with initial data, boundary data and right hand side function from the whole scale of Lebesgue‐type spaces and prove that the problem has a unique solution I . This paper uses the technique of Amosov,() which, in contrast to other studies,() allows to prove the existence of a solution without the additional assumption of existence of a derivative of initial value I 0 . Note that the results of Amosov() in the case under consideration are not applicable, because the verical layers are unbounded in .…”

“…1 In recent years, considerable attention has been paid to the study of boundary value problems for the stationary radiation transfer equation with the conditions of reflection and refraction in different statements and under various constraints on the structure of the domain. [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32] Similar problems for the nonstationary radiation transfer equation are studied in much less detail 33,34 (1D case), [35][36][37][38] (3D case).…”