Inverse Problem of Determining the Absorption Coefficient in a Degenerate Parabolic Equation in the Class of L 2 -Functi
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Journal of Mathematical Sciences, Vol. 250, No. 2, October, 2020
INVERSE PROBLEM OF DETERMINING THE ABSORPTION COEFFICIENT IN A DEGENERATE PARABOLIC EQUATION IN THE CLASS OF L2 -FUNCTIONS V. L. Kamynin National Research Nuclear University MEPhI 31, Kashirskoe shosse, Moscow 115409, Russia [email protected]
UDC 517.95
We study the unique solvability of the inverse problem of determining the time-dependent lower order coefficient in a degenerate parabolic equation with one spatial variable under an additional integral observation condition. The unknown absorption coefficient is sought in the space L2 (0, T ). We prove the time-local existence and global uniqueness of a solution to the inverse problem. Bibliography: 17 titles.
1
Introduction
We study the unique solvability of the inverse problem of finding an unknown coefficient γ(t) ∈ L2 (0, T ) in the parabolic equation ut − a(t, x)uxx + b(t, x)ux + c(t, x)u + γ(t)u = f (t, x),
(t, x) ∈ Q,
(1.1)
with the initial and boundary conditions u(0, x) = u0 (x),
x ∈ [0, l],
u(t, 0) = u(t, l) = 0,
t ∈ [0, T ],
(1.2) (1.3)
and the additional integral observation condition l u(t, x)ω(x)dx = ϕ(t),
t ∈ [0, T ].
(1.4)
0
Here, Q = [0, T ] × [0, l], T, l are some numbers, a(t, x), b(t, x), c(t, x), f (t, x), u0 (x), ω(x), ϕ(t) are known functions. Moreover, the equation can be degenerate 0 a(t, x) a1 ,
1/a(t, x) ∈ Lq (Q), q > 1.
Translated from Problemy Matematicheskogo Analiza 105, 2020, pp. 121-133. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2502-0322
322
(1.5)
For uniformly parabolic equations the inverse problems of determining the absorption coefficient γ(t) under the integral observation condition (1.4) are studied in a lot of works (cf., for example, [1]–[6]). In particular, the inverse problem for the mutlidimensional second order equation with the principal part in divergence form was studied in [3, 4] in the class of bounded functions. In [5] and [6], the inverse problem was studied in the class of functions in L2 (0, T ) for higher order equations in nondivergence form with one spatial variable and for second order equations in nondivergence form with many spatial variables respectively. For degenerate parabolic equations of the form (1.1) with the condition (1.5) the inverse problems of determining the time-dependent source on the right-hand side of the equation was studied by the author in [7, 8]. We also mention the works [9, 10] devoted to the inverse problems of determining the unknown x-dependent source on the right-hand side of a nonuniformly parabolic equation under conditions of type (1.5). Finally, the inverse problems of determining the unknown lower order coefficient γ(t) in the parabolic equation (1.1) under the integral observation condition (1.4) were studied in [11, 12], but in the case where the higher order coefficient a(t, x) of Equation (1.1) is strongly degenerate at x = 0: instead of (1.5), it was assumed that a(t, x) ∼ a0 xα , x → 0, α = const 2. In both works [11, 12], the coefficient γ(t) was looked for in the
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