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On the statement and solvability of general parabolic boundary value conjugation problems in spaces of generalized functions. (Russian) Zbl 0569.35044

The following adjoint problem is considered: find the solution of the system \[ {\mathcal L}_{(m)}(x,t,D_ x,D_ t)u_{(m)}(x,t)=f_{(m)}(x,t),\quad (x,t)\in \Omega_ m,\quad m=1,2,..., \] which satisfies the boundary adjoint condition \[ {\mathcal B}_{(1)}(x,t,D_ x,D_ t)u_ 1(s,t)|_ S+{\mathcal B}_{(2)}(x,t,D_ x,D_ t)u_{(2)}(x,t)|_ S=\phi (x',t) \] and for \(t=0\), the initial condition \[ C_{(m)}(x,t,D_ x,D_ t)u_{(m)}(x,t)|_{t=0}=\psi_{(m)}(x),\quad m=1,2,.... \] A vector function \(u_{(m)}(x,t)\) with the components belonging to a Sobolev-Slobodzinski type space is called a generalized solution for the adjoint problem if some additional equalities - Green formulas in distribution theory - are satisfied. The uniqueness of such a generalized solution is discussed.
Reviewer: C.Simionescu

MSC:

35K35 Initial-boundary value problems for higher-order parabolic equations
47F05 General theory of partial differential operators
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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