Mathematics & Physics 2015, 8(3), 281–290 УДК 517.95 On some Inverse Problem for a Parabolic Equation with a Parameter Kirill V.Korshun∗ Institute of Mathematics and Computer Science Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041 Russia Received 10.05.2015, received in revised form 01.06.2015, accepted 20.06.2015 An inverse boundary-value problem for n-dimensional parabolic equation with a parameter is considered. <...> Sufficient conditions for existence and uniqueness of solution in continuously differentiable class are obtained. <...> Coefficient inverse problems for parabolic equations are problems of finding solutions of differential equation with one (or more) unknown coefficients. <...> An inverse problem for a parabolic equation with a parameter is investigated. <...> Inverse problems with unknown parameter arise in various problems: in studying boundaryvalue problems for mixed-type equations and equation systems [2, 3]; in solving various inverse problems [4–7]; in studying boundary-value problems for equation systems with small parameters [8,9]. 1. <...> Then the problem has a solution of class Zp. <...> The solution of problem (1)–(4) of class Zp is unique. <...> This problem has a solution of class Zp(Rn) if conditions (5) are fulfilled in domain E. b. <...> Proof of existence The proof of Theorem 1.1 is based on reduction of boundary-value problem to Cauchy problem. <...> We construct an extension of functions u0, f from set QT to E in n steps. <...> At i-th step (2 i n) we extend functions u0, f from [0, li] to R with respect to variable xi in the same way. <...> If we assume that u∗ 0, f∗ are arbitrary functions satisfying (5) in domain E then we prove Theorem 1.3 a. <...> We assume τ → 0 and xi = 0 in (22)–(23) and Kirill V.Korshun On some Inverse Problem for a Parabolic Equation with a Paramete 3. <...> Kirill V.Korshun On some Inverse Problem for a Parabolic Equation with a Paramete This proves that the right-hand side of (28) is equal to zero. <...> Appendix A. Proof of statement (21) Split-problem (18)–(20) is n-dimensional Cauchy problem for parabolic equation (18), (20) at the first fractional step and the Cauchy problem for ordinary differential equation (19), (20) at the second fractional <...>