After the change z = στ , we obtain |v2yy (y, σ)| ≤ C36 σ
γ −γ 2β
1 0
1−β
−γ
γ z β dz −γ 2β ≤ C σ . 37 (1 − z 1/β )1−γ/2
γ − γ > −1. Therefore, the singularity of v2yy (y, σ) is integrable. Then, We note that γ ∈ (0, 1) yields 2β due to properties of second-order Volterra integral equations, the system of equations (4.5–(4.8) has only the trivial solution v(y, σ) ≡ 0, h(σ) ≡ 0, a(σ) ≡ 0, p(σ) ≡ 0, y ∈ [0, σ], σ ∈ [0, T1 ]. Therefore, the solution of problem (1.5)–(1.8) and, hence, that of problem (1.1–(1.4) are unique. The theorem is proved.
Theorem 4.2. Let β > 1, and let the condition (A6) ˜b, c˜, f˜ ∈ C 1,0 ([0, ∞) × [0, T ]), µi ∈ C 1 [0, T ], µ i (t) = λi (t)tβ−1 , λi ∈ C[0, T ], i = 1, 2, µi (t) = 0, i = 2, 3, µ4 (t) = µ0 (t)tβ , µ0 (t) = 0, t ∈ [0, T ] ˜ be satisfied. Then problem (1.1)–(1.4) cannot have more than one solution (h(t), a ˜(t), u(x, t)), which 1 2,1 1,0 belongs to the class C (0, T ] ∩ C[0, T ] × C[0, T ] × C (QT ) ∩ C (QT ). Proof. The proofs of Theorems 4.1 and 4.2 differ from each other only by the estimates of the expression σ
1−β β
v2yy (y, σ). We now fix some γ ∈ (0, 1) and estimate the expression σ τ σ ν2 (τ ) − ν1 (τ ) dτ dτ G∗∗ η dη ≤ C38 , yy (y, σ, η, τ ) γ τ τ (θ2 (σ) − θ2 (τ ))1−γ/2 0
0
(4.14)
0
taking into account that condition (A6) yields νi (σ) =
1 1/β 1−β 1 µ (σ )σ β = λi (σ 1/β ), β i β
i = 1, 2.
Then relations (4.12) and (4.14) yield σ |v2yy (y, σ)| ≤ C39 0
dτ ≤ C40 τ γ (θ2 (σ) − θ2 (τ ))1−γ/2
σ 0
τ γ (σ 1/β
γ dτ 1− β1 + 2β −γ ≤ C σ . 41 − τ 1/β )1−γ/2
Using this inequality, we find the estimate |σ
1−β β
γ
v2yy (y, σ)| ≤ C42 σ 2β
−γ
,
and the proof of Theorem 4.2 is completed analogously to that of Theorem 4.1. REFERENCES 1. I. Barans’ka, “Inverse problem for a parabolic equation in a free-boundary domain,” Mat. Met. Fiz.-Mekh. Polya, 48, No. 2, 32–42 (2005). 2. I. Barans’ka and M. Ivanchov, “Inverse problem for a two-dimensional parabolic equation in a free-boundary domain,” Ukr. Mat. Visn., 4, No. 4, 467–484 (2007). 3. A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, 1964.
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