Abstract
In the present article we examine an inverse problem of recovering unknown functions being part of the Dirichlet boundary condition together solving an initial boundary problem for a parabolic second order equation. Such problems on recovering the boundary data arise in various tasks of mathematical physics: control of heat exchange prosesses and design of thermal protection systems, diagnostics and identification of heat transfer in supersonic heterogeneous flows, identification and modeling of heat transfer in heat-shielding materials and coatings, modeling of properties and heat regimes of reusable heat protection of spacecrafts, study of composite materials, etc. As the overdetrermination conditions we take the integrals of a solution over the spatial domain with weights. The problem is reduced to an operator equation of the Volterra-type. The existence and uniqueness theorem for solutions to this inverse problem is established in Sobolev spaces. A solution is regular, i. e., all generalized derivatives occuring into the equation exists and are summable to some power. The proof relies on the fixed point theorem and bootstrap arguments. Stability estimates for solutions are also given. The solvability conditions are close to necessary conditions
Highlights
We consider the parabolic equation ∑ ∑ Lu = ut n − ∂i, j=1 ∂xi aij (t, x)ux j +n ai (t, x)uxi i =1 + a0 (t, x)u = f
+ a0 (t, x)u where x ∈G ⊂ Rn is a bounded domain with boundary Γ of the class C2, t ∈ (0,T )
The overdetrermination conditions are the values of a solution at some points lying inside the spatial domain
Summary
Where x ∈G ⊂ Rn is a bounded domain with boundary Γ of the class C2 (see the definition, for instance, in [1, Sect. 1]), t ∈ (0,T ). The overdetrermination conditions are the values of a solution at some points lying inside the spatial domain These problems are studied in different settings in dependence on the type of the ovedetermination conditions. It is often the case when these problems are ill-posed in the Hadamard sense It is true in the case of the overdetermination conditions in the form of values of a solution at separate points or on some surfaces lying in the spatial domain (see [2]). At the present article we examine the problems with overdetermination conditions in the form of some integrals with weights of a solution over a spatial domain. We employ the Holder spaces (see the definition for instance, in [22]) Cα ,β (Q),Cα ,β (S) , Ck (G) and the Sobolev spaces
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: Bulletin of the South Ural State University series "Mathematics. Mechanics. Physics"
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
