Page 1

Journal of Mathematical Sciences, Vol. 181, No. 1, February, 2012

An inverse problem for a parabolic equation in a free-boundary domain degenerating at the initial time moment Mykola I. Ivanchov and Tetyana Savitska Presented by A. E. Shishkov

Abstract. We consider an inverse problem for a one-dimensional parabolic equation with unknown time-dependent major coefficient in a domain whose unknown boundary weakly degenerates at the initial time moment. The conditions for existence and uniqueness of the classical solution of the problem are established. Keywords. Parabolic equation, inverse problem, free boundary with degeneration.

Introduction The study of many practically important processes is associated with the solution of inverse problems for partial differential equations in domains with free boundaries. Some types of such problems were considered in [1, 2, 6, 13]. The study of these problems in separate cases is complicated by the degeneration of the equation or the free boundary [5]. Inverse problems for parabolic equations with degeneration were investigated, in particular, in [9–11]. Analogous problems in free-boundary domains were studied in [4]. The boundary-value problem for the heat equation in a domain with free boundary degenerating at the initial time moment was considered in [7]. In the present work, we study the inverse problem for a one-dimensional parabolic equation with the unknown time-dependent major coefficient in a domain with free boundary degenerating at the initial time moment. We will show that this problem in the case of power degeneration of the boundary can be reduced to an equation with power degeneration in the known triangular domain. In the case of weak degeneration, we establish the conditions for the existence and the uniqueness of a solution.

1.

Statement of the problem

˜ In the domain QT ≡ {(x, t) : 0 < x < tβ h(t), 0 < t < T } we consider the inverse problem for the parabolic equation ˜(t)uxx + ˜b(x, t)ux + c˜(x, t)u + f˜(x, t) (1.1) ut = a under the conditions u(0, t) = µ1 (t),

˜ u(tβ h(t), t) = µ2 (t),

a ˜(t)ux (0, t) = µ3 (t),

t ∈ [0, T ],

t ∈ [0, T ],

(1.2) (1.3)

˜ tβh(t)

u(x, t) dx = µ4 (t),

t ∈ [0, T ],

(1.4)

0

Translated from Ukrains’ki˘ı Matematychny˘ı Visnyk, Vol. 8, No. 3, pp. 381–403, July–August, 2011. Original article submitted December 15, 2010 c 2012 Springer Science+Business Media, Inc. 1072 – 3374/12/1811–0047 

47


˜ ˜ where β > 0 is a given number, and (h(t), a ˜(t), u(x, t)), a ˜(t) > 0, h(t) > 0, t ∈ [0, T ] are the unknowns. x β By the change y = h(t) ˜ , σ = t , we reduce problem (1.1)–(1.4) to the inverse problem for the parabolic equation with degeneration: 1−β

1−β

σ β a(σ) βyh (σ) + σ β b(yh(σ), σ) vσ = v vy + yy βh2 (σ) βh(σ) σ

+

1−β β



β

 c(yh(σ), σ)v + f (yh(σ), σ) ,

(y, σ) ∈ QT1 , (1.5)

v(σ, σ) = ν2 (σ), σ ∈ [0, T1 ],

v(0, σ) = ν1 (σ),

(1.6)

a(σ)vy (0, σ) = h(σ)ν3 (σ), σ ∈ [0, T1 ], σ h(σ) v(y, σ) dy = ν4 (σ), σ ∈ [0, T1 ].

(1.7) (1.8)

0 1

1

˜ β ), σ β ), a(σ) = Here, QT1 = {(y, σ) : 0 < y < σ, 0 < σ < T1 }, T1 = T β , v(y, σ) = u(y h(σ 1 1 1 1 1 1 ˜ β ), b(yh(σ), σ) = ˜b(y h(σ ˜ β ), σ β ), c(yh(σ), σ) = c˜(y h(σ ˜ β ), σ β ), f (yh(σ), σ) = a ˜(σ β ), h(σ) = h(σ 1 1 1 ˜ β ), σ β ), νi (σ) = µi (σ β ), i = 1, 4. It is obvious that problems (1.1)–(1.4) and (1.5)–(1.8) are f˜(y h(σ equivalent. We note that, in the case where β > 12 , the degeneration of Eq. (1.5) is weak. Just this case will be considered in what follows.

2.

Boundary-value problem for the heat equation in a triangular domain Consider the following auxiliary problem: ut = a(t)tγ uxx + f (x, t), u(0, t) = µ1 (t),

0 < x < t, 0 < t < T,

u(t, t) = µ2 (t),

t ∈ [0, T ].

(2.1) (2.2)

If 0 < γ < 1, a ∈ C[0, T ], a(t) > 0, t ∈ [0, T ], µi ∈ C[0, T ], i = 1, 2, µ1 (0) = µ2 (0), f ∈ C([0, t] × [0, T ]), and the function f (x, t) satisfies the H¨older condition in x, then the solution of problem (2.1), (2.2) from the class C 2,1 ((0, t) × (0, T )) ∩ C([0, t] × [0, T ]) can be presented in the form [8] t

t Gξ (x, t, 0, τ )a(τ )µ1 (τ ) dτ −

u(x, t) = 0

Gξ (x, t, τ, τ )a(τ )µ2 (τ ) dτ 0

t τ G(x, t, ξ, τ )f (ξ, τ ) dξdτ, (2.3)

+ 0

0

where G(x, t, ξ, τ ) is the Green function of problem (2.1), (2.2), which can be found in the form t G(x, t, ξ, τ ) = G0 (x, t, ξ, τ ) +

dσ τ

48

σ G0 (x, t, η, σ)Φ(η, σ, ξ, τ ) dη. 0


Here, ∞  

1

G0 (x, t, ξ, τ ) =  2 π(θ(t) − θ(τ )) n=−∞

  (x − ξ + 2nt)2 exp − − 4(θ(t) − θ(τ ))   (x + ξ + 2nt)2 − exp − , 4(θ(t) − θ(τ ))

t a(τ )τ γ dτ,

θ(t) = 0

and the function Φ(x, t, ξ, τ ) is the solution of the equation t Φ(x, t, ξ, τ ) = −LG0 (x, t, ξ, τ ) −

σ dσ

τ

in which L =

3.

∂ ∂t

LG0 (x, t, η, σ)Φ(η, σ, ξ, τ ) dη 0

2

∂ − a(t)tγ ∂x 2.

Existence of a solution of problem (1.1)–(1.4)

We perform the study of the existence of a solution of problem (1.1)–(1.4) separately in the cases where 12 < β ≤ 1 and β > 1. Theorem 3.1. At

1 2

< β ≤ 1, let the following assumptions be valid:

(A1) µi ∈ C 1 [0, T ], i = 1, 2, 4, µ3 ∈ C[0, T ], µi (t) = λi (t)tβ−1 , λi ∈ C[0, T ], i = 1, 2; the functions ˜b, c˜, f˜ ∈ C([0, ∞) × [0, T ]) satisfy locally the H¨ older condition in x with exponent α ∈ (0, 1) uniformly relative to t ∈ [0, T ]; (A2) µi (t) > 0, i = 1, 2, 3, µ4 (t) = µ0 (t)tβ , µ0 (t) > 0, t ∈ [0, T ], f˜(x, t) ≥ 0, c˜(x, t) ≤ 0, (x, t) ∈ [0, ∞) × [0, T ], λ2 (t) > λ1 (t), t ∈ [0, T ]; (A3) µ1 (0) = µ2 (0). ˜ Then there exists at least one solution (h(t), a ˜(t), u(x, t)) of problem (1.1)–(1.4), which is defined at ˜ 0 ≤ x ≤ h(t), 0 ≤ t ≤ T0 , and belongs to the class C 1 (0, T0 ]∩C[0, T0 ]×C[0, T0 ]×C 2,1 (QT0 )∩C 1,0 (QT0 ), where the number T0 , 0 < T0 ≤ T is determined by the initial data of the problem. Proof. Due to the equivalence of problems (1.1)–(1.4) and (1.5)–(1.8), it is sufficient to prove the existence of a solution of the last problem. Changing the unknown function v(y, σ) = v˜(y, σ) + ν1 (σ) + y σ (ν2 (σ) − ν1 (σ)), we reduce this problem to the following one: 1−β

1−β

σ β a(σ) βyh (σ) + σ β b(yh(σ), σ) v ˜ v˜y + v˜σ = yy βh2 (σ) βh(σ) +

σ

1−β β

β



 y c(yh(σ), σ)˜ v + f (yh(σ), σ) − ν1 (σ) − (ν2 (σ) − ν1 (σ)) σ

(ν2 (σ) − ν1 (σ))(βyh (σ) + σ y + 2 (ν2 (σ) − ν1 (σ)) + σ βσh(σ) +

σ

1−β β

b(yh(σ), σ))

1−β β

  y c(yh(σ), σ) ν1 (σ) + (ν2 (σ) − ν1 (σ)) , (y, σ) ∈ QT1 , (3.1) β σ 49


v˜(0, σ) = v˜(σ, σ) = 0,

σ ∈ [0, T1 ].

(3.2)

Using the Green function of the problem 1−β

σ β a(σ) vσ = vyy , (y, σ) ∈ QT1 , βh2 (σ) v(0, σ) = v(σ, σ) = 0, σ ∈ [0, T1 ],

(3.3) (3.4)

we reduce problem (3.1), (3.2) to the integro-differential equation σ v˜(y, σ) =

τ dτ

0

0

+

1−β  βηh (τ ) + τ β b(ηh(τ ), τ ) v˜η (η, τ ) G(y, σ, η, τ ) βh(τ )

τ

1−β β



β

 η c(ηh(τ ), τ )˜ v (η, τ ) + f (ηh(τ ), τ ) − ν1 (τ ) − (ν2 (τ ) − ν1 (τ )) τ 1−β

η (ν2 (τ ) − ν1 (τ ))(βηh (τ ) + τ β b(ηh(τ ), τ )) + 2 (ν2 (τ ) − ν1 (τ )) + τ βτ h(τ ) 1−β   τ β η c(ηh(τ ), τ ) ν1 (τ ) + (ν2 (τ ) − ν1 (τ )) dη, + β τ

(y, σ) ∈ QT1 .

Returning to the function v(y, σ), we obtain v(y, σ) = ν1 (σ) +

y (ν2 (σ) − ν1 (σ)) σ 1−β  σ τ βηh (τ ) + τ β b(ηh(τ ), τ ) + dτ G(y, σ, η, τ ) vη (η, τ ) βh(τ ) 0

+

τ

0

1−β β

  η c(ηh(τ ), τ )v(η, τ ) + f (ηh(τ ), τ ) − ν1 (τ ) − (ν2 (τ ) β τ  η  − ν1 (τ )) + 2 (ν2 (τ ) − ν1 (τ )) dη, (y, σ) ∈ QT1 . (3.5) τ

Let us find the lower bound for the function v(y, σ). By v0 (y, σ), we denote the solution of the problem 1−β

1−β

1−β

σ β a(σ) βyh (σ) + σ β b(yh(σ), σ) σ β vyy + vy + c(yh(σ), σ)v, vσ = 2 βh (σ) βh(σ) β v(0, σ) = ν1 (σ),

v(σ, σ) = ν2 (σ),

(y, σ) ∈ QT1 ,

0 ≤ σ ≤ T1 ,

(3.6) (3.7)

and, by v (y, σ), the solution of the problem 1−β

1−β

σ β a(σ) βyh (σ) + σ β b(yh(σ), σ) vσ = vyy + vy 2 βh (σ) βh(σ) + 50

σ

1−β β

β

  c(yh(σ), σ)v + f (yh(σ), σ) ,

(y, σ) ∈ QT1 , (3.8)


0 â&#x2030;¤ Ď&#x192; â&#x2030;¤ T1 .

v(0, Ď&#x192;) = v(Ď&#x192;, Ď&#x192;) = 0,

(3.9)

Then v(y, Ď&#x192;) = v0 (y, Ď&#x192;) + v (y, Ď&#x192;). From the maximum principle [3, Section 2], we obtain

v0 (y, Ď&#x192;) â&#x2030;Ľ C1 min min Âľ1 (t), min Âľ2 (t) = M1 > 0, [0,T ]

[0,T ]

(3.10)

(y, Ď&#x192;) â&#x2C6;&#x2C6; QT1 .

Since the conditions of the theorem imply that v (y, Ď&#x192;) â&#x2030;Ľ 0, we have v(y, Ď&#x192;) â&#x2030;Ľ M1 > 0,

(y, Ď&#x192;) â&#x2C6;&#x2C6; QT1 .

(3.11)

With regard for this result, we represent Eq. (1.8) in the form h(Ď&#x192;) =

Ď&#x192; 0

ν4 (Ď&#x192;)

,

Ď&#x192; â&#x2C6;&#x2C6; (0, T1 ].

(3.12)

v(y, Ď&#x192;) dy

We denote w(y, Ď&#x192;) â&#x2030;Ą vy (y, Ď&#x192;), p(Ď&#x192;) â&#x2030;Ą Ď&#x192;h (Ď&#x192;) and reduce Eq. (3.5) to the system of integral equations v(y, Ď&#x192;) = ν1 (Ď&#x192;) +

y (ν2 (Ď&#x192;) â&#x2C6;&#x2019; ν1 (Ď&#x192;)) Ď&#x192; 1â&#x2C6;&#x2019;β   Ď&#x192; Ď&#x201E; Ρp(Ď&#x201E; ) Ď&#x201E; β b(Ρh(Ď&#x201E; ), Ď&#x201E; ) + + dĎ&#x201E; G(y, Ď&#x192;, Ρ, Ď&#x201E; ) w(Ρ, Ď&#x201E; ) Ď&#x201E; h(Ď&#x201E; ) βh(Ď&#x201E; ) 0

+

Ď&#x201E;

0

1â&#x2C6;&#x2019;β β

  Ρ c(Ρh(Ď&#x201E; ), Ď&#x201E; )v(Ρ, Ď&#x201E; ) + f (Ρh(Ď&#x201E; ), Ď&#x201E; ) â&#x2C6;&#x2019; ν1 (Ď&#x201E; ) â&#x2C6;&#x2019; (ν2 (Ď&#x201E; ) β Ď&#x201E;  Ρ  â&#x2C6;&#x2019; ν1 (Ď&#x201E; )) + 2 (ν2 (Ď&#x201E; ) â&#x2C6;&#x2019; ν1 (Ď&#x201E; )) dΡ, (y, Ď&#x192;) â&#x2C6;&#x2C6; QT1 , (3.13) Ď&#x201E;

ν2 (Ď&#x192;) â&#x2C6;&#x2019; ν1 (Ď&#x192;) + w(y, Ď&#x192;) = Ď&#x192; +

Ď&#x201E;

Ď&#x192;

Ď&#x201E; dĎ&#x201E;

0

0

1â&#x2C6;&#x2019;β   Ρp(Ď&#x201E; ) Ď&#x201E; β b(Ρh(Ď&#x201E; ), Ď&#x201E; ) + w(Ρ, Ď&#x201E; ) Gy (y, Ď&#x192;, Ρ, Ď&#x201E; ) Ď&#x201E; h(Ď&#x201E; ) βh(Ď&#x201E; )

1â&#x2C6;&#x2019;β β

  Ρ c(Ρh(Ď&#x201E; ), Ď&#x201E; )v(Ρ, Ď&#x201E; ) + f (Ρh(Ď&#x201E; ), Ď&#x201E; ) â&#x2C6;&#x2019; ν1 (Ď&#x201E; ) â&#x2C6;&#x2019; (ν2 (Ď&#x201E; ) β Ď&#x201E;  Ρ  â&#x2C6;&#x2019; ν1 (Ď&#x201E; )) + 2 (ν2 (Ď&#x201E; ) â&#x2C6;&#x2019; ν1 (Ď&#x201E; )) dΡ, (y, Ď&#x192;) â&#x2C6;&#x2C6; QT1 . (3.14) Ď&#x201E;

Condition (1.7) yields the equation a(Ď&#x192;)w(0, Ď&#x192;) = h(Ď&#x192;)ν3 (Ď&#x192;),

Ď&#x192; â&#x2C6;&#x2C6; [0, T1 ].

(3.15)

DiďŹ&#x20AC;erentiating condition (1.8) and using (1.5), we obtain the equation  1â&#x2C6;&#x2019;β 1 Ď&#x192; β a(Ď&#x192;)  ν (Ď&#x192;) â&#x2C6;&#x2019; h(Ď&#x192;)ν2 (Ď&#x192;) â&#x2C6;&#x2019; (w(Ď&#x192;, Ď&#x192;) â&#x2C6;&#x2019; w(0, Ď&#x192;)) p(Ď&#x192;) = ν2 (Ď&#x192;) 4 βh(Ď&#x192;) 51


â&#x2C6;&#x2019;

Ď&#x192;

1â&#x2C6;&#x2019;β β

β

Ď&#x192;

   b(yh(Ď&#x192;), Ď&#x192;)w(y, Ď&#x192;) + h(Ď&#x192;)(c(yh(Ď&#x192;), Ď&#x192;)v(y, Ď&#x192;) + f (yh(Ď&#x192;), Ď&#x192;)) dy ,

Ď&#x192; â&#x2C6;&#x2C6; (0, T1 ]. (3.16)

0

Hence, problem (1.5)â&#x20AC;&#x201C;(1.8) is reduced to the system of integral equations (3.12)â&#x20AC;&#x201C;(3.16) for the unknowns (h(Ď&#x192;), a(Ď&#x192;), v(y, Ď&#x192;), w(y, Ď&#x192;), and p(Ď&#x192;)). Repeating the reasoning in [2], it is easy to verify that problem (1.5)â&#x20AC;&#x201C;(1.8) is equivalent to the problem of the determination of a continuous solution of the system of integral equations (3.12)â&#x20AC;&#x201C;(3.16). Let us establish the existence of a continuous solution of the system of integral equations (3.12)â&#x20AC;&#x201C; (3.16), by using the Schauder theorem on ďŹ xed point for a completely continuous operator. For this purpose, we will obtain the estimates of solutions of the system of equations (3.12)â&#x20AC;&#x201C;(3.16). Relations (3.11) and (3.12) yield the inequality h(Ď&#x192;) â&#x2030;¤ H1 < â&#x2C6;&#x17E;,

Ď&#x192; â&#x2C6;&#x2C6; [0, T1 ].

(3.17)

Using this estimate and the maximum principle [3, Section 2], we have v(y, Ď&#x192;) â&#x2030;¤ M2 < â&#x2C6;&#x17E;,

(y, Ď&#x192;) â&#x2C6;&#x2C6; QT1 .

(3.18)

In turn, relations (3.12) and (3.18) and condition (A2) yield h(Ď&#x192;) â&#x2030;Ľ

Ď&#x192;ν0 (Ď&#x192;) â&#x2030;Ľ H0 > 0, Ď&#x192;M2

Ď&#x192; â&#x2C6;&#x2C6; [0, T1 ],

(3.19)

where ν0 (Ď&#x192;) = Âľ0 (Ď&#x192; 1/β ). It follows from conditions (A1)â&#x20AC;&#x201C;(A3) that lim

Ď&#x192;â&#x2020;&#x2019;0

1 ν2 (Ď&#x192;) â&#x2C6;&#x2019; ν1 (Ď&#x192;) = ν2 (0) â&#x2C6;&#x2019; ν1 (0) = (Îť2 (0) â&#x2C6;&#x2019; Îť1 (0)) > 0, Ď&#x192; β

Ď&#x192; â&#x2C6;&#x2C6; (0, T1 ].

Therefore,

ν2 (Ď&#x192;) â&#x2C6;&#x2019; ν1 (Ď&#x192;) â&#x2030;Ľ M3 > 0, Ď&#x192; â&#x2C6;&#x2C6; (0, T1 ]. (3.20) Ď&#x192; Let us ďŹ nd the behavior of the integral term in formula (3.14). We assume that the quantities w(y, Ď&#x192;), a(Ď&#x192;), and p(Ď&#x192;) are bounded. In view of the estimates of the Green function [12, c. 469], we have Ď&#x192; Ď&#x201E; 1â&#x2C6;&#x2019;β    Ρp(Ď&#x201E; ) Ď&#x201E; β b(Ρh(Ď&#x201E; ), Ď&#x201E; )  + w(Ρ, Ď&#x201E; )  dĎ&#x201E; Gy (y, Ď&#x192;, Ρ, Ď&#x201E; )  Ď&#x201E; h(Ď&#x201E; ) βh(Ď&#x201E; ) 0 0 1â&#x2C6;&#x2019;β    Ρ  Ρ Ď&#x201E; β     + c(Ρh(Ď&#x201E; ), Ď&#x201E; )v(Ρ, Ď&#x201E; ) + f (Ρh(Ď&#x201E; ), Ď&#x201E; ) â&#x2C6;&#x2019; ν1 (Ď&#x201E; ) â&#x2C6;&#x2019; (ν2 (Ď&#x201E; ) â&#x2C6;&#x2019; ν1 (Ď&#x201E; )) + 2 (ν2 (Ď&#x201E; ) â&#x2C6;&#x2019; ν1 (Ď&#x201E; )) dΡ   β Ď&#x201E; Ď&#x201E; Ď&#x192; â&#x2030;¤ C8

Ď&#x201E; dĎ&#x201E;

0

  Gy (y, Ď&#x192;, Ρ, Ď&#x201E; )dΡ â&#x2030;¤ C9

0

 0

Making change z = Ď&#x192; 0

52

Ď&#x192;

Ď&#x201E; Ď&#x192;,

dĎ&#x201E; θ(Ď&#x192;) â&#x2C6;&#x2019; θ(Ď&#x201E; )

Ď&#x192; â&#x2030;¤ C10 0

dĎ&#x201E;  , 1/β Ď&#x192; â&#x2C6;&#x2019; Ď&#x201E; 1/β

(y, Ď&#x192;) â&#x2C6;&#x2C6; QT1 . (3.21)

we obtain dĎ&#x201E;

 =Ď&#x192; Ď&#x192; 1/β â&#x2C6;&#x2019; Ď&#x201E; 1/β

1 1â&#x2C6;&#x2019; 2β

1 0

dz 1â&#x2C6;&#x2019; 1  â&#x2030;¤ C11 Ď&#x192; 2β , 1 â&#x2C6;&#x2019; z 1/β

(y, Ď&#x192;) â&#x2C6;&#x2C6; QT1 .

(3.22)


Since estimate (3.20) is valid for the first term in (3.14), there exists a number σ1 , 0 < σ1 ≤ T1 , such that, for y ∈ [0, σ], σ ∈ [0, σ1 ], the estimate 1 w(y, σ) ≥ M3 > 0 2

(3.23)

holds. Then relations (3.15) and (3.17) yield a(σ) ≤ A1 < ∞,

σ ∈ [0, σ1 ].

(3.24)

We denote amin (σ) ≡ min a(τ ).

W (σ) ≡ max w(y, σ), 0≤y≤σ

0≤τ ≤σ

Relations (3.14) and (3.16) yield the inequalities σ W (σ) ≤ C12 + C13 0

(1 + |p(τ )|)(W (τ ) + 1)  dτ, θ(σ) − θ(τ )

σ ∈ [0, σ1 ],

σ ∈ [0, σ1 ].

|p(σ)| ≤ C14 + C15 W (σ),

(3.25) (3.26)

Substituting (3.26) in (3.25), we obtain σ W (σ) ≤ C12 + C16 0

(W (τ ) + 1)2  dτ. θ(σ) − θ(τ )

After the change W1 (σ) ≡ W (σ) + 1, the obtained inequality takes the form σ

C18

W1 (σ) ≤ C17 +  amin (σ) Since τ ≤ σ and β ≤ 1, we have

 τ 1/β−1 σ

0

W 2 (τ ) dτ  1 . σ 1/β − τ 1/β

(3.27)

≤ 1, and inequality (3.27) is reduced to σ

C18

W1 (σ) ≤ C17 +  amin (σ)σ 1/β−1

0

W12 (τ ) dτ √ . σ−τ

(3.28)

Let us square this inequality. Using the Cauchy and Cauchy–Buniakowski inequalities, we obtain W12 (σ)

C20 ≤ C19 + amin (σ)σ 1/β−3/2

We set σ = s in the previous inequality, multiply by σ 0

√ W12 (s) ds √ ≤ C21 σ + C20 σ−s

σ 0

ds amin (s)s1/β−3/2

≤ C21

s 0

C20 σ+ amin (σ)

√1 , σ−s

σ 0

W14 (τ ) dτ √ . σ−τ

and integrate from 0 to σ:

W 4 (τ ) dτ  1 (s − τ )(σ − s) σ 0

W14 (τ ) dτ

σ τ

s1/β−3/2

ds  . (3.29) (s − τ )(σ − s) 53


Let

1 2

< β < 23 . Then, in view of the equality Ď&#x192; Ď&#x201E;

we have

Ď&#x192; 0

ds  = Ď&#x20AC;, (s â&#x2C6;&#x2019; Ď&#x201E; )(Ď&#x192; â&#x2C6;&#x2019; s)

â&#x2C6;&#x161; C22 W12 (s) ds â&#x2C6;&#x161; â&#x2030;¤ C21 Ď&#x192; + amin (Ď&#x192;) Ď&#x192;â&#x2C6;&#x2019;s

Ď&#x192; 0

W14 (Ď&#x201E; ) dĎ&#x201E; . Ď&#x201E; 1/βâ&#x2C6;&#x2019;3/2

Substituting the last inequality in (3.28), we obtain 2βâ&#x2C6;&#x2019;1

C24 C23 Ď&#x192; 2β + W1 (Ď&#x192;) â&#x2030;¤ C17 +  1â&#x2C6;&#x2019;β 3/2 amin (Ď&#x192;) a (Ď&#x192;)Ď&#x192; 2β min

Ď&#x192; 0

W14 (Ď&#x201E; ) dĎ&#x201E; Ď&#x201E; 1/βâ&#x2C6;&#x2019;3/2 2βâ&#x2C6;&#x2019;1

C24 C23 Ď&#x192; 2β + 3/2 â&#x2030;¤ C17 +  amin (Ď&#x192;) amin (Ď&#x192;)

Ď&#x192;

W14 (Ď&#x201E; ) dĎ&#x201E; 3

0

Ď&#x201E; 2β

â&#x2C6;&#x2019;2

. (3.30)

We denote 2βâ&#x2C6;&#x2019;1

C24 C23 Ď&#x192; 2β , ÎŚ(Ď&#x192;) = C17 +  , Ψ(Ď&#x192;) = 3/2 amin (Ď&#x192;) amin (Ď&#x192;) Ď&#x192; 4 W1 (Ď&#x201E; ) dĎ&#x201E; ÎŚ(Ď&#x192;) + . H(Ď&#x192;) = 3 â&#x2C6;&#x2019;2 Ψ(Ď&#x192;) 2β Ď&#x201E; 0

(3.31)

(3.32)

Then relation (3.30) yields W1 (Ď&#x192;) â&#x2030;¤ H(Ď&#x192;). Ψ(Ď&#x192;)

(3.33)

DiďŹ&#x20AC;erentiating (3.32) with respect to Ď&#x192; and taking (3.33) into account, we obtain 

H (Ď&#x192;) â&#x2030;¤



ÎŚ(Ď&#x192;) Ψ(Ď&#x192;)

 +

H 4 (Ď&#x192;)Ψ4 (Ď&#x192;) 3

Ď&#x192; 2β

â&#x2C6;&#x2019;2

.

(3.34)

We now divide this inequality by H 4 (Ď&#x192;) and integrate from 0 to Ď&#x192;: 1 ÎŚ(Ď&#x192;) ÎŚ(0) 1 â&#x2C6;&#x2019; â&#x2030;¤ â&#x2C6;&#x2019; +4 3 3 4 3H (0) 3H (Ď&#x192;) Ψ(Ď&#x192;)H (Ď&#x192;) Ψ(0)H 4 (0)

It follows from (3.32) that H(0) =

Ď&#x192; 0

ÎŚ(Ď&#x201E; )H  (Ď&#x201E; ) dĎ&#x201E; + Ψ(Ď&#x201E; )H 5 (Ď&#x201E; )

Ď&#x192;

Ψ4 (Ď&#x201E; ) dĎ&#x201E; 3

0

Ď&#x201E; 2β

â&#x2C6;&#x2019;2

Ό(0) . Hence, Ψ(0)

   Ď&#x192; Ď&#x192; 4 H 4 (Ď&#x192;) ÎŚ(Ď&#x192;) H(Ď&#x192;) ÎŚ(Ď&#x201E; )H  (Ď&#x201E; ) Ψ (Ď&#x201E; ) dĎ&#x201E; 3 â&#x2030;¤ 4 â&#x2C6;&#x2019; 3H (0) 4 dĎ&#x201E; + + . 3 â&#x2C6;&#x2019;2 3H 3 (0) Ψ(Ď&#x201E; )H 5 (Ď&#x201E; ) Ψ(Ď&#x192;) 3 2β Ď&#x201E; 0 0 54

.

(3.35)


Making change z = H(Ď&#x201E; ) in the integral Ď&#x192; 0

Ď&#x192;

ÎŚ(Ď&#x201E; )H  (Ď&#x201E; ) 0 Ψ(Ď&#x201E; )H 5 (Ď&#x201E; )

ÎŚ(Ď&#x201E; )H  (Ď&#x201E; ) dĎ&#x201E; = Ψ(Ď&#x201E; )H 5 (Ď&#x201E; )

dĎ&#x201E;, we obtain

H(Ď&#x192;) 

ÎŚ(H â&#x2C6;&#x2019;1 (z)) dz , Ψ(H â&#x2C6;&#x2019;1 (z))z 5

H(0)

where H â&#x2C6;&#x2019;1 (z) is the function inverse to H(Ď&#x192;). Since H(Ď&#x192;)  Ď&#x192; 4 ÎŚ(H â&#x2C6;&#x2019;1 (z)) dz Ψ (Ď&#x201E; ) dĎ&#x201E; 4 + â&#x2020;&#x2019; 0 at Ď&#x192; â&#x2020;&#x2019; 0, 3 â&#x2C6;&#x2019;2 Ψ(H â&#x2C6;&#x2019;1 (z))z 5 2β Ď&#x201E; 0 H(0)

there exists a number Ď&#x192;2 : 0 < Ď&#x192;2 â&#x2030;¤ T1 such that   Ď&#x192; Ď&#x192; 4  (Ď&#x201E; ) ÎŚ(Ď&#x201E; )H Ψ (Ď&#x201E; ) dĎ&#x201E; 4 â&#x2C6;&#x2019; 3H 3 (0) 4 dĎ&#x201E; + â&#x2030;Ľ 1, 3 â&#x2C6;&#x2019;2 Ψ(Ď&#x201E; )H 5 (Ď&#x201E; ) 2β Ď&#x201E; 0 0

Ď&#x192; â&#x2C6;&#x2C6; [0, Ď&#x192;2 ].

(3.36)

Then relation (3.35) yields the inequality ÎŚ(Ď&#x192;) H(Ď&#x192;) H 4 (Ď&#x192;) â&#x2030;¤ + , 3 3H (0) Ψ(Ď&#x192;) 3 or H 4 (Ď&#x192;) â&#x2030;¤

Ď&#x192; â&#x2C6;&#x2C6; [0, Ď&#x192;2 ],

3ÎŚ(Ď&#x192;) 3 H (0) + H(Ď&#x192;)H 3 (0), Ψ(Ď&#x192;)

Ď&#x192; â&#x2C6;&#x2C6; [0, Ď&#x192;2 ].

Applying this result to (3.34), we ďŹ nd   ÎŚ(Ď&#x192;)  3ÎŚ(Ď&#x192;)Ψ3 (Ď&#x192;)H 3 (0) H(Ď&#x192;)H 3 (0)Ψ4 (Ď&#x192;)  + + . H (Ď&#x192;) â&#x2030;¤ 3 3 â&#x2C6;&#x2019;2 â&#x2C6;&#x2019;2 Ψ(Ď&#x192;) Ď&#x192; 2β Ď&#x192; 2β   Ď&#x192; 4 We multiply this inequality by exp â&#x2C6;&#x2019;H 3 (0) 0 Ψ 3(s)â&#x2C6;&#x2019;2ds and integrate from 0 to Ď&#x192;: s 2β

 3

Ď&#x192;

H(Ď&#x192;) exp â&#x2C6;&#x2019;H (0)

Ψ4 (s) ds 3

0

s 2β

â&#x2C6;&#x2019;2

 â&#x2C6;&#x2019; H(0)   Ď&#x192; 4 Ψ (s) ds ÎŚ(Ď&#x192;) exp â&#x2C6;&#x2019;H 3 (0) â&#x2030;¤ 3 â&#x2C6;&#x2019;2 Ψ(Ď&#x192;) s 2β 0 â&#x2C6;&#x2019;

Ό(0) + 4H 3 (0) Ψ(0)

Ď&#x192;

 Ď&#x201E; ÎŚ(Ď&#x201E; )Ψ3 (Ď&#x201E; ) exp â&#x2C6;&#x2019;H 3 (0) 3

Ď&#x201E; 2β

0

0

â&#x2C6;&#x2019;2

Ψ4 (s) ds



3 â&#x2C6;&#x2019;2

s 2β

dĎ&#x201E;.

The last inequality can be presented in the form ÎŚ(Ď&#x192;) H(Ď&#x192;) â&#x2030;¤ + 4H 3 (0) Ψ(Ď&#x192;)

Ď&#x192;

ÎŚ(Ď&#x201E; )Ψ3 (Ď&#x201E; ) exp

 Ď&#x192; H 3 (0) 3

0

Ď&#x201E; 2β

Ď&#x201E; â&#x2C6;&#x2019;2

Ψ4 (s) ds



3 â&#x2C6;&#x2019;2

s 2β

dĎ&#x201E;.

55


Then relation (3.33) yields W1 (Ď&#x192;) â&#x2030;¤ ÎŚ(Ď&#x192;) + 4H 3 (0)Ψ(Ď&#x192;)

Ď&#x192;

ÎŚ(Ď&#x201E; )Ψ3 (Ď&#x201E; ) dĎ&#x201E; 3

Ď&#x201E; 2β

0

â&#x2C6;&#x2019;2

  Ď&#x192; 4 Ψ (s) ds 3 . exp H (0) 3 â&#x2C6;&#x2019;2 2β s 0

In view of DeďŹ nition (3.31), we obtain 2βâ&#x2C6;&#x2019;1

C25 C23 Ď&#x192; 2β + 3/2 W (Ď&#x192;) â&#x2030;¤ C17 +  amin (Ď&#x192;) amin (Ď&#x192;)

2βâ&#x2C6;&#x2019;1  Ď&#x192;  dĎ&#x201E; C23 Ď&#x201E; 2β C17 +  3 9/2 amin (Ď&#x201E; ) a (Ď&#x201E; )Ď&#x201E; 2β â&#x2C6;&#x2019;2

min

0



Ď&#x192;

Ă&#x2014; exp C26



ds 3

0

a6min (s)s 2β

â&#x2C6;&#x2019;2

.

There exists a number Ď&#x192;3 â&#x2C6;&#x2C6; (0, T1 ] such that   2βâ&#x2C6;&#x2019;1  Ď&#x192;  Ď&#x192; dĎ&#x201E; C23 Ď&#x201E; 2β ds C17 +  â&#x2030;¤ C17 . exp C26 3 3 3/2 6 (s)s 2β â&#x2C6;&#x2019;2 amin (Ď&#x201E; ) a9/2 (Ď&#x201E; )Ď&#x201E; 2β â&#x2C6;&#x2019;2 amin (Ď&#x192;) a min min 0 0 C25

Therefore,

(3.37)

2βâ&#x2C6;&#x2019;1

C23 Ď&#x192; 2β . W (Ď&#x192;) â&#x2030;¤ 2C17 +  amin (Ď&#x192;)

(3.38)

This result and relation (3.15) yield the inequality 2C17 amin (Ď&#x192;) + C23 Ď&#x192; i.e.,  amin (Ď&#x192;) â&#x2030;Ľ Hence, amin (Ď&#x192;) â&#x2030;Ľ

2βâ&#x2C6;&#x2019;1 2β

 amin (Ď&#x192;) â&#x2C6;&#x2019; C27 â&#x2030;Ľ 0,

 2βâ&#x2C6;&#x2019;1 2βâ&#x2C6;&#x2019;1 2 Ď&#x192; β + 8C C â&#x2C6;&#x2019; C Ď&#x192; 2β C23 17 27 23 . 4C17

2  2βâ&#x2C6;&#x2019;1 2βâ&#x2C6;&#x2019;1 2 β 2β C23 Ď&#x192; + 8C17 C27 â&#x2C6;&#x2019; C23 Ď&#x192; .

2 16C17

(3.39)

Relations (3.39), (3.38), and (3.16) yield a(Ď&#x192;) â&#x2030;Ľ A0 > 0, |p(Ď&#x192;)| â&#x2030;¤ M4 , w(y, Ď&#x192;) â&#x2030;¤ M5 ,

Ď&#x192; â&#x2C6;&#x2C6; [0, Ď&#x192;â&#x2C6;&#x2014; ],

(y, Ď&#x192;) â&#x2C6;&#x2C6; QĎ&#x192;â&#x2C6;&#x2014; ,

(3.40)

where Ď&#x192;â&#x2C6;&#x2014; = min{Ď&#x192;2 , Ď&#x192;3 }, and the constants A0 , M4 , and M5 depend only on the initial data. It is easy to see that estimates (3.40) hold also in the case where 23 â&#x2030;¤ β â&#x2030;¤ 1, since inequality (3.29) 3/2â&#x2C6;&#x2019;1/β

. Thus, we obtain the estimates of solutions of can be extended with regard for s3/2â&#x2C6;&#x2019;1/β â&#x2030;¤ T1 system (3.12)â&#x20AC;&#x201C;(3.16). The system of equations (3.12)â&#x20AC;&#x201C;(3.16) can be presented as Ď&#x2030; = P Ď&#x2030;, 56

(3.41)


where ω = (h, v, w, a, p), and the operator P is determined by the right-hand sides of Eqs. (3.12)– (3.16). In this case, Eqs. (3.15) should be rewritten in the form a(σ) =

h(σ)ν3 (σ) , w(0, σ)

σ ∈ (0, T1 ].

Using the obtained estimates of solutions of the system of equations (3.12)–(3.16) and applying the procedure given in [2], we determine the set N ≡ {(h, v, w, a, p) ∈ C[0, σ0 ] × (C(Qσ0 ))2 × (C[0, σ0 ])2 : H0 ≤ h(σ) ≤ H1 , M6 ≤ v ≤ M7 , M8 ≤ w ≤ M9 , |p| ≤ M10 }, where the number σ0 , 0 < σ0 ≤ T, is determined by the initial data so that the operator P transfers the set N into itself. The compactness of the operator P on the set N was established in [14]. Using the Schauder theorem, we obtain that there exists at least one fixed point of the operator P in N . This means that there exists the classical solution of problem (1.1)–(1.4). The theorem is proved. We note that, in the case β = 1, Eq. (1.5) does not degenerate. Let β > 1, and let the condition (A4) µ4 (t) ≡ tβ−1 λ4 (t), f˜(x, t) ≡ tβ−1 f0 (x, t), ˜b(x, t) ≡ tβ−1 b0 (x, t), c˜(x, t) ≡ tβ−1 c0 (x, t) be satisfied. Here, the functions f0 , b0 , c0 ∈ C([0, +∞) × [0, T ]) satisfy locally the H¨older condition in x with exponent α ∈ (0, 1), and the function λ4 (t) is continuous on [0, T ]. Theorem 3.2. Let β > 1, and let assumptions (A1)–(A4) be satisfied. Then there exists at least ˜ one solution (h(t), a ˜(t), u(x, t)) of problem (1.1)–(1.4), which belongs to the class C 1 (0, T0 ] ∩ C[0, T0 ] × C[0, T0 ]×C 2,1 (QT0 )∩C 1,0 (QT0 ), where T0 , 0 < T0 ≤ T is determined by the initial data of the problem. Proof. The proofs of Theorems 3.1 and 3.2 differ from each other only by estimates of the expressions 1−β

1

1−β

(w(σ, σ) − w(0, σ))σ β−1 , σ β b(yh(σ), σ)w(y, σ), σ β (f (yh(σ), σ) + c(yh(σ), σ)v(y, σ)). By condition (A4), two last expressions are bounded. With regard for (3.14), we consider the expression σ

w(σ, σ) − w(0, σ) =

dτ 0

+

(Gy (σ, σ, η, τ ) 0

1−β   ηp(τ ) τ β b(ηh(τ ), τ ) + w(η, τ ) − Gy (0, σ, η, τ )) τ h(τ ) βh(τ )

τ

1−β β

  η c(ηh(τ ), τ )v(η, τ ) + f (ηh(τ ), τ ) − ν1 (τ ) − (ν2 (τ ) β τ  η  − ν1 (τ )) + 2 (ν2 (τ ) − ν1 (τ )) dη, τ

(y, σ) ∈ QT1 .

Since the functions G(σ, σ, η, τ ) and G0 (σ, σ, η, τ ) have the same singularities, it is sufficient to estimate the expression σ τ   (1) (1) dτ G0y (σ, σ, η, τ ) − G0y (0, σ, η, τ ) dη, 0

0

where we used the notation (i) G0 (y, σ, η, τ )

  ∞   (y − η + 2nσ)2 1  exp − = 4(θ(σ) − θ(τ )) 2 π(θ(σ) − θ(τ )) n=−∞   (y + η + 2nσ)2 i , + (−1) exp − 4(θ(σ) − θ(τ ))

i = 1, 2, 57


a(τ )

θ(σ) = 0

βτ

β−1 β

h2 (τ )

dτ.

Taking into account that (1)

G0y (0, σ, η, τ ) ≥ 0, (1)

(1)

G0y (σ, σ, η, τ ) ≤ 0, (2)

G0y (y, σ, η, τ ) = −G0η (y, σ, η, τ ), we have σ

0

(1)

(1)

|G0y (σ, σ, η, τ ) − G0y (0, σ, η, τ )| dη

dτ 0

σ 0

(2)

(2)

(G0η (σ, σ, η, τ ) − G0η (0, σ, η, τ )) dη

= σ

τ 0

(2)

(2)

(2)

(2)

(G0 (σ, σ, τ, τ ) − G0 (0, σ, τ, τ ) − G0 (σ, σ, 0, τ ) + G0 (0, σ, 0, τ )) dτ

= 0

  ∞   ((2n + 1)σ − τ )2  exp − = 4(θ(σ) − θ(τ )) 2 π(θ(σ) − θ(τ )) n=−∞ 0     (2nσ − τ )2 ((2n + 1)σ + τ )2 − exp − + exp − 4(θ(σ) − θ(τ )) 4(θ(σ) − θ(τ ))       2 2 ((2n + 1)σ) (2nσ)2 (2nσ + τ ) − 2 exp − + 2 exp − dτ − exp − 4(θ(σ) − θ(τ )) 4(θ(σ) − θ(τ )) 4(θ(σ) − θ(τ ))    σ ∞  (nσ − τ )2 1 n+1  = exp − (−1) 4(θ(σ) − θ(τ )) 2 π(θ(σ) − θ(τ )) n=−∞ 0     (nσ)2 (nσ + τ )2 − 2 exp − dτ + exp − 4(θ(σ) − θ(τ )) 4(θ(σ) − θ(τ ))      σ ∞  (nσ)2 (nσ + τ )2 1 n  = − exp − dτ. (−1) exp − 4(θ(σ) − θ(τ )) 4(θ(σ) − θ(τ )) π(θ(σ) − θ(τ )) n=−∞ σ

1

0

We note that 

 exp −

 (nσ + τ )2 (−1) 4(θ(σ) − θ(τ )) n=−∞    (nσ + τ )2 (nσ)2 + exp − 4(θ(σ) − θ(τ )) 4(θ(σ) − θ(τ )) n=−∞        ∞  (nσ)2 (nσ + τ )2 (nσ − τ )2 2 exp − + exp − + exp − . ≤2+ 4(θ(σ) − θ(τ )) 4(θ(σ) − θ(τ )) 4(θ(σ) − θ(τ )) ∞ 

n

n=1

58

(nσ)2 4(θ(σ) − θ(τ ))  ∞   exp − ≤



 − exp −


Using Lemma 2.1.1 in [8], we obtain σ

τ dτ

0

 (1)  G (σ, σ, η, τ ) − G(1) (0, σ, η, τ ) dη ≤ C28 0y 0y

0

σ 0

or

dτ  + C29 σ θ(σ) − θ(τ ) 1

|(w(σ, σ) − w(0, σ))σ 1/β−1 | ≤ C30 σ 2β .

(3.42)

Due to estimate (3.42), the proof of Theorem 3.2 is completed in the same way as that of Theorem 3.1.

4.

Uniqueness of a solution of problem (1.1)–(1.4)

Theorem 4.1. Let

1 2

< β ≤ 1, and let the condition

(A5) ˜b, c˜, f˜ ∈ C 1,0 ([0, +∞) × [0, T ]), µi ∈ C 1 [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. Since problem (1.1)–(1.4) is equivalent to problem (1.5)–(1.8), it is sufficient to prove the uniqueness of the solution of problem (1.5)–(1.8). We assume that there exist two different solutions (hi (σ), ai (σ), vi (y, σ)), i = 1, 2 of problem (1.5)–(1.8). We denote h(σ) ≡ h1 (σ) − h2 (σ), a(σ) ≡ a1 (σ) − a2 (σ), v(y, σ) ≡ v1 (y, σ) − v2 (y, σ). Let us transform the difference 1 b(yh1 (σ), σ) − b(yh2 (σ), σ) = y(h1 (σ) − h2 (σ)) 0

 ∂b(z, σ)  ds. ∂z z=yh2 (σ)+sy(h1 (σ)−h2 (σ))

Analogously, we can present the differences c(yh1 (σ), σ) − c(yh2 (σ), σ) and f (yh1 (σ), σ) − f (yh2 (σ), σ). Then relations (1.5)–(1.8) yield the following problem for (h(σ), a(σ), v(y, σ)): vσ =

σ

1−β  σ β c(yh1 (σ), σ) b(yh1 (σ), σ) vy + v βh1 (σ) β 1−β  y(h (σ)h2 (σ) − h2 (σ)h(σ)) σ β (a(σ)h22 (σ) − a2 (σ)h(σ)(h1 (σ) + h2 (σ))) v + + 2yy h1 (σ)h2 (σ) βh21 (σ)h22 (σ) 1−β    1 ∂b(z, σ)  σ β yh(σ)h2 (σ) ds − b(yh2 (σ), σ)h(σ) v2y + βh1 (σ)h2 (σ) ∂z z=yh2 (σ)+syh(σ)

1−β β

a1 (σ) vyy + βh21 (σ)



yh1 (σ) σ + h1 (σ)

1−β β

0

  1 σ ∂c(z, σ)  h(σ)yv2 + ds β ∂z z=yh2 (σ)+syh(σ) 1−β β

0

1 + yh(σ) 0

  ∂f (z, σ)  ds , ∂z z=yh2 (σ)+syh(σ)

(y, σ) ∈ QT1 , (4.1)

59


v(0, Ď&#x192;) = v(Ď&#x192;, Ď&#x192;) = 0,

Ď&#x192; â&#x2C6;&#x2C6; [0, T1 ],

a1 (Ď&#x192;)vy (0, Ď&#x192;) = ν3 (Ď&#x192;)h(Ď&#x192;) â&#x2C6;&#x2019; a(Ď&#x192;)v2y (0, Ď&#x192;), Ď&#x192; 0

Using the Green function vĎ&#x192; =

Ď&#x192;

Ď&#x192; â&#x2C6;&#x2C6; [0, T1 ],

(4.3)

Ď&#x192; â&#x2C6;&#x2C6; [0, T1 ].

(4.4)

Ď&#x192; v(y, Ď&#x192;) dy = â&#x2C6;&#x2019;h(Ď&#x192;)

h1 (Ď&#x192;)

(4.2)

v2 (y, Ď&#x192;) dy, 0

Gâ&#x2C6;&#x2014; (y, Ď&#x192;, Ρ, Ď&#x201E; )

1â&#x2C6;&#x2019;β β

a1 (Ď&#x192;) vyy + 2 βh1 (Ď&#x192;)



of the ďŹ rst boundary-value problem for the equation

yh1 (Ď&#x192;) Ď&#x192; + h1 (Ď&#x192;)

1â&#x2C6;&#x2019;β β

 b(yh1 (Ď&#x192;), Ď&#x192;) vy , βh1 (Ď&#x192;)

(y, Ď&#x192;) â&#x2C6;&#x2C6; QT1 ,

we reduce problem (4.1)â&#x20AC;&#x201C;(4.4) to the system of equations Ď&#x192;

Ď&#x201E; dĎ&#x201E;

v(y, Ď&#x192;) = 0

0

 1â&#x2C6;&#x2019;β Ď&#x201E; β c(Ρh1 (Ď&#x201E; ), Ď&#x201E; ) G (y, Ď&#x192;, Ρ, Ď&#x201E; ) v(Ρ, Ď&#x201E; ) β â&#x2C6;&#x2014;

1â&#x2C6;&#x2019;β β

(a(Ď&#x201E; )h22 (Ď&#x201E; ) â&#x2C6;&#x2019; a2 (Ď&#x201E; )h(Ď&#x201E; )(h1 (Ď&#x201E; ) + h2 (Ď&#x201E; ))) v2ΡΡ (Ρ, Ď&#x201E; ) βh21 (Ď&#x201E; )h22 (Ď&#x201E; ) 1â&#x2C6;&#x2019;β   Ď&#x201E; β Ρ(h (Ď&#x201E; )h2 (Ď&#x201E; ) â&#x2C6;&#x2019; h2 (Ď&#x201E; )h(Ď&#x201E; )) + Ρh(Ď&#x201E; )h2 (Ď&#x201E; ) + h1 (Ď&#x201E; )h2 (Ď&#x201E; ) βh1 (Ď&#x201E; )h2 (Ď&#x201E; )   1 â&#x2C6;&#x201A;b(z, Ď&#x201E; )  ds â&#x2C6;&#x2019; b(Ρh2 (Ď&#x201E; ), Ď&#x201E; )h(Ď&#x201E; ) v2Ρ (Ρ, Ď&#x201E; ) Ă&#x2014; â&#x2C6;&#x201A;z z=yh2 (Ď&#x201E; )+syh(Ď&#x201E; ) +

Ď&#x201E;

0

+

Ď&#x201E;

  1 â&#x2C6;&#x201A;c(z, Ď&#x201E; )  ds h(Ď&#x201E; )Ρv2 (Ρ, Ď&#x201E; ) β â&#x2C6;&#x201A;z z=yh2 (Ď&#x201E; )+syh(Ď&#x201E; )

1â&#x2C6;&#x2019;β β

0

1 + Ρh(Ď&#x201E; ) 0

h1 (Ď&#x192;) h(Ď&#x192;) = â&#x2C6;&#x2019;

Ď&#x192; 0

a(Ď&#x192;) =

Ď&#x192; 0

  â&#x2C6;&#x201A;f (z, Ď&#x201E; )  ds dΡ, â&#x2C6;&#x201A;z z=yh2 (Ď&#x201E; )+syh(Ď&#x201E; ) v(y, Ď&#x192;)dy ,

Ď&#x192; â&#x2C6;&#x2C6; [0, T1 ],

ν3 (Ď&#x192;)h(Ď&#x192;) â&#x2C6;&#x2019; a1 (Ď&#x192;)vy (0, Ď&#x192;) , v2y (0, Ď&#x192;)

Ď&#x192; â&#x2C6;&#x2C6; [0, T1 ],

0

60

(4.6)

v2 (y, Ď&#x192;)dy

 1â&#x2C6;&#x2019;β  a1 (Ď&#x192;) Ď&#x192; β 1 â&#x2C6;&#x2019;h(Ď&#x192;)ν2 (Ď&#x192;) â&#x2C6;&#x2019; (vy (Ď&#x192;, Ď&#x192;) â&#x2C6;&#x2019; vy (0, Ď&#x192;)) p(Ď&#x192;) = ν2 (Ď&#x192;) β h1 (Ď&#x192;)  Ď&#x192;  b(yh1 (Ď&#x192;), Ď&#x192;)vy (y, Ď&#x192;) + h1 (Ď&#x192;)c(yh1 (Ď&#x192;), Ď&#x192;)v(y, Ď&#x192;) dy + +

(y, Ď&#x192;) â&#x2C6;&#x2C6; QT1 , (4.5)

a(Ď&#x192;)h2 (Ď&#x192;) â&#x2C6;&#x2019; a2 (Ď&#x192;)h(Ď&#x192;) (v2y (Ď&#x192;, Ď&#x192;) â&#x2C6;&#x2019; v2y (0, Ď&#x192;)) h1 (Ď&#x192;)h2 (Ď&#x192;)

(4.7)


 Ď&#x192;  1 â&#x2C6;&#x201A;b(z, Ď&#x192;)  yh(Ď&#x192;)v2y (y, Ď&#x192;) + ds â&#x2C6;&#x201A;z z=yh2 (Ď&#x192;)+syh(Ď&#x192;) 0

1 + (h1 (Ď&#x192;)yh(Ď&#x192;) 0

 â&#x2C6;&#x201A;c(z, Ď&#x192;)  â&#x2C6;&#x201A;z 

0

z=yh2 (Ď&#x192;)+syh(Ď&#x192;)

1 + yh(Ď&#x192;) 0

ds + h(Ď&#x192;)c(yh2 (Ď&#x192;), Ď&#x192;))v2 (y, Ď&#x192;)

   â&#x2C6;&#x201A;f (z, Ď&#x192;)  ds dy , â&#x2C6;&#x201A;z z=yh2 (Ď&#x192;)+syh(Ď&#x192;)

Ď&#x192; â&#x2C6;&#x2C6; [0, T1 ]. (4.8)

By substituting (4.5) in formula (4.6) and the result together with (4.5) in (4.7) and (4.8), it is easy to verify that relations (4.5)â&#x20AC;&#x201C;(4.8) are the system of second-order Volterra integral equations. Let us establish the integrability of the kernels of system (4.5)â&#x20AC;&#x201C;(4.8). Condition (A5) and the fact that v2 (y, Ď&#x192;) satisďŹ es (1.7) and (1.8) imply that the denominators in (4.7) and (4.8) are nonzero. Since v2 (y, Ď&#x192;) satisďŹ es (1.8), relation (4.6) and condition (A5) yield h1 (Ď&#x192;)h2 (Ď&#x192;) h(Ď&#x192;) = â&#x2C6;&#x2019;

Ď&#x192; 0

v(y, Ď&#x192;) dy

ν4 (Ď&#x192;) h1 (Ď&#x192;)h2 (Ď&#x192;) =â&#x2C6;&#x2019;

Ď&#x192; 0

v(y, Ď&#x192;) dy

h1 (Ď&#x192;)h2 (Ď&#x192;)Ď&#x192;v(Ë&#x153; y , Ď&#x192;) Ď&#x192;ν0 (Ď&#x192;) y , Ď&#x192;) h1 (Ď&#x192;)h2 (Ď&#x192;)v(Ë&#x153; , yË&#x153; â&#x2C6;&#x2C6; [0, Ď&#x192;], Ď&#x192; â&#x2C6;&#x2C6; [0, T1 ]. =â&#x2C6;&#x2019; ν0 (Ď&#x192;) =â&#x2C6;&#x2019;

Ď&#x192;ν0 (Ď&#x192;)

Hence, the kernel in (4.6) is integrable. Let us study the behavior of v2yy (y, Ď&#x192;) as Ď&#x192; â&#x2020;&#x2019; 0. We make change v2 (y, Ď&#x192;) = vË&#x153;2 (y, Ď&#x192;) + ν1 (Ď&#x192;) + y (ν Ë&#x153;2 (y, Ď&#x192;): Ď&#x192; 2 (Ď&#x192;) â&#x2C6;&#x2019; ν1 (Ď&#x192;)). Then relations (1.5)â&#x20AC;&#x201C;(1.8) yield the problem for the function v vË&#x153;2Ď&#x192; =

Ď&#x192;

1â&#x2C6;&#x2019;β β

a2 (Ď&#x192;) vË&#x153;2yy + 2 βh2 (Ď&#x192;) 1â&#x2C6;&#x2019;β β



yh2 (Ď&#x192;) Ď&#x192; + h2 (Ď&#x192;)

1â&#x2C6;&#x2019;β β

 b(yh2 (Ď&#x192;), Ď&#x192;) vË&#x153;2y βh2 (Ď&#x192;)



 y c(yh2 (Ď&#x192;), Ď&#x192;)Ë&#x153; v2 + f (yh2 (Ď&#x192;), Ď&#x192;) â&#x2C6;&#x2019; ν1 (Ď&#x192;) â&#x2C6;&#x2019; (ν2 (Ď&#x192;) â&#x2C6;&#x2019; ν1 (Ď&#x192;)) β Ď&#x192; 1â&#x2C6;&#x2019;β    yh2 (Ď&#x192;) Ď&#x192; β b(yh2 (Ď&#x192;), Ď&#x192;) ν2 (Ď&#x192;) â&#x2C6;&#x2019; ν1 (Ď&#x192;) y + 2 (ν2 (Ď&#x192;) â&#x2C6;&#x2019; ν1 (Ď&#x192;)) + + Ď&#x192; h2 (Ď&#x192;) βh2 (Ď&#x192;) Ď&#x192; 1â&#x2C6;&#x2019;β   Ď&#x192; β y + c(yh2 (Ď&#x192;), Ď&#x192;) ν1 (Ď&#x192;) + (ν2 (Ď&#x192;) â&#x2C6;&#x2019; ν1 (Ď&#x192;)) , (y, Ď&#x192;) â&#x2C6;&#x2C6; QT1 , (4.9) β Ď&#x192; +

Ď&#x192;

vË&#x153;2 (0, Ď&#x192;) = vË&#x153;2 (Ď&#x192;, Ď&#x192;) = 0,

Ď&#x192; â&#x2C6;&#x2C6; [0, T1 ].

(4.10)

Using the Green function Gâ&#x2C6;&#x2014;â&#x2C6;&#x2014; (y, Ď&#x192;, Ρ, Ď&#x201E; ) of the ďŹ rst boundary-value problem for the equation vË&#x153;2Ď&#x192; =

Ď&#x192;

1â&#x2C6;&#x2019;β β

a2 (Ď&#x192;) vË&#x153;2yy + 2 βh2 (Ď&#x192;)



yh2 (Ď&#x192;) Ď&#x192; + h2 (Ď&#x192;)

1â&#x2C6;&#x2019;β β

1â&#x2C6;&#x2019;β  b(yh2 (Ď&#x192;), Ď&#x192;) Ď&#x192; β vË&#x153;2y + c(yh2 (Ď&#x192;), Ď&#x192;)Ë&#x153; v2 , βh2 (Ď&#x192;) β

we obtain the solution of problem (4.9), (4.10), whence we have

61


v2 (y, σ) = ν1 (σ) +

y (ν2 (σ) − ν1 (σ)) σ  1−β σ τ τ β ∗∗ f (ηh2 (τ ), τ ) − ν1 (τ ) + dτ G (y, σ, η, τ ) β 0

0

η η − (ν2 (τ ) − ν1 (τ )) + 2 (ν2 (τ ) − ν1 (τ ))+ τ τ 1−β    β ηh2 (τ ) τ b(ηh2 (τ ), τ ) ν2 (τ ) − ν1 (τ ) + + h2 (τ ) βh2 (τ ) τ 1−β   η τ β c(ηh2 (τ ), τ ) ν1 (τ ) + (ν2 (τ ) − ν1 (τ )) dη, + β τ

(y, σ) ∈ QT1 . (4.11)

Let us differentiate formula (4.11) twice with respect to y: σ v2yy (y, σ) =

τ dτ

0



G∗∗ yy (y, σ, η, τ )

0

τ

1−β β

β

f (ηh2 (τ ), τ )

η η − ν1 (τ ) − (ν2 (τ ) − ν1 (τ )) + 2 (ν2 (τ ) − ν1 (τ )) τ τ 1−β    ηh2 (τ ) τ β b(ηh2 (τ ), τ ) ν2 (τ ) − ν1 (τ ) + + h2 (τ ) βh2 (τ ) τ 1−β   τ β η c(ηh2 (τ ), τ ) ν1 (τ ) + (ν2 (τ ) − ν1 (τ )) dη, + β τ

(y, σ) ∈ QT1 . (4.12)

To estimate this result, we use the estimate of second derivatives of the bulk heat potentials [3] in the form   σ τ σ   dτ   ∗∗ .  dτ Gyy (y, σ, η, τ )f (η, τ ) dη  ≤ C31   (θ2 (σ) − θ2 (τ ))1−γ/2 0

0

0

Here, f (y, σ) is the function continuous in QT , which satisfies the H¨older condition in the variable y with exponent γ, 0 < γ < 1. We represent η as η = η γ η 1−γ . In view of the fact that the function η γ satisfies the H¨older condition with exponent γ, 0 < γ < 1, and the equality νi (σ) = β1 µi (σ 1/β )σ 1/β−1 , i = 1, 2, holds, relation (4.12) yields   σ τ 1−β σ −γ    (τ ) − ν  (τ ) ν τ β dτ   ∗∗ 2 1 η dη  ≤ C32 , (4.13)  dτ Gyy (y, σ, η, τ )   τ (θ2 (σ) − θ2 (τ ))1−γ/2 0

0

0

where

σ θ2 (σ) = 0

1−β

a2 (τ )τ β dτ. βh22 (τ )

The remaining terms in (4.12) can be estimated analogously, using condition (A5). With regard for the inequality θ2 (σ) − θ2 (τ ) ≥ C33 (σ 1/β − τ 1/β ), relations (4.12) and (4.13) yield σ |v2yy (y, σ)| ≤ C34 0

62

1−β

−γ

τ β dτ ≤ C35 (θ(σ) − θ(τ ))1−γ/2

σ 0

τ (σ 1/β

1−β −γ β

τ 1/β )1−γ/2

.


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 ], µi (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.

63


4. N. Hryntsiv, “Inverse problem for a parabolic equation with strong power degeneration in a free-boundary domain,” Visn. Lviv. Univ. Ser. Mekh.-Mat., Issue 64, 84–97 (2007). 5. V. Isakov, “On inverse problems in secondary oil recovery,” Eur. J. of Appl. Math., 19, 459–478 (2008). 6. M. I. Ivanchov, “Inverse problem with free boundary for the heat equation,” Ukr. Mat. Zh., 55, No. 7, 901–910 (2003). 7. M. I. Ivanchov, “The problem of heat conduction with free boundary degenerating at the initial time moment,” Mat. Met. Fiz.-Mekh. Polya, 50, No. 3, 82–87 (2007). 8. M. Ivanchov, Inverse Problem for Equation of Parabolic Type, VNTL, Lviv, 2003. 9. M. Ivanchov and N. Saldina, “An inverse problem for strongly degenerate heat equation,” J. Inv. Ill-Posed Probl., 14, No. 5, 465–480 (2006). 10. M. Ivanchov, A. Lorenzi, and N. Saldina, “Solving a scalar degenerate multidimensional identification problem in a Banach space,” J. Inv. Ill-Posed Probl., 16, No. 4, 397–415 (2008). 11. M. Ivanchov and N. Saldina, “An inverse problem for a parabolic equation with strong power degeneration,” Ukr. Mat. Zh., 58, No. 11, 1487–1500 (2006). 12. O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence, RI, 1968. 13. L. Lorenzi, “An identification problem for a one-phase Stefan problem,” J. Inv. Ill-Posed Probl., 9, No. 6, 1–27 (2001). 14. G. A. Snitko, “Inverse problem for a parabolic equation in a free-boundary domain,” Mat. Met. Fiz.-Mekh. Polya, 50, No. 4, 7–18 (2007). Translated from Ukrainian by V. V. Kukhtin

Mykola Ivanovych Ivanchov and Tetyana Savitska I. Franko Lviv National University 1, Universytets’ka Str., Lviv 79001, Ukraine E-Mail: ivanchov@franko.lviv.ua

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