GEORGIAN MATHEMATICAL JOURNAL: Vol. 5, No. 4, 1998, 321-332

THE DEPENDENCE OF SOLUTION UNIQUENESS CLASSES OF BOUNDARY VALUE PROBLEMS FOR

GENERAL PARABOLIC SYSTEMS ON THE GEOMETRY OF AN UNBOUNDED DOMAIN

A. GAGNIDZE

Abstract. General boundary value problems are considered for general parabolic (in the Douglas–Nirenberg–Solonnikov sense) systems. The dependence of solution uniqueness classes of these problems on the geometry of a nonbounded domain is established.

The dependence of solution uniqueness classes of the first boundary value problem for a second-order parabolic equation in an unbounded domain on the domain geometry was considered in [2]. It was established there that the uniqueness class could be wider than the solution uniqueness class of the Cauchy problem for the above-mentioned equation. Analogous results were obtained in [2] by the method of barrier functions.

In [3] O. Oleinik constructed examples of second-order parabolic equations in the exponentially narrowing domains for which the solution uniqueness class for the second boundary value problem is the same as the solution uniqueness class of the Cauchy problem although the solution uniqueness class in this domain is wider. Later, in [4] E. Landis showed that if the domain narrows with a sufficient quickness at |x| → ∞, then the solution uniqueness class for the second boundary value problem can be wider than that of the Cauchy problem. In [5] the author considered degenerating parabolic equations of second order and obtained analogous uniqueness theorems for general boundary value problems.

This paper deals with general boundary value problems for general parabolic systems in unbounded domains. Such problems were studied in [6], where solvability conditions similar to the Shapiro-Lopatinski conditions for

321 elliptic boundary value problems were found. The solvability conditions for analogous problems in unbounded domains were obtained in [7], [8].

1991 Mathematics Subject Classification. 35K50.

Key words and phrases. Parabolic systems, uniqueness classes, influence of domain geometry.

1072-947X/98/0700-0321$15.00/0 \begin{matrix}c \\ ◯ \end{matrix} 1998 Plenum Publishing Corporation

Let ω be an unbounded domain in ℝ^{n+1} contained between the planes {t = 0} and {t = T = const > 0} and the surface γ lying in-between these planes. Let us consider, in this domain, a linear system of differential equations with complex-valued coefficients of the form

∑_{j=1}^{N} ∑_{|α|+2bβ≤s_{k}+t_{j}} a_{kj}^{αβ}(x, t)D_{x}^{α} \frac{∂^{β}}{∂t^{β}} u_{j}(x, t) = 0,  (1)

where b, N are positive integers, s_{1}, s_{2}, …, s_{N}, t_{1}, t_{2}, …, t_{N}, β are integers, s_{j} ≤ 0 and t_{j} ≥ 0 for all j = 1, 2, …, N, a_{kj}^{αβ}(x, t) ≡ 0 if s_{k} + t_{j} < 0;

∑_{j=1}^{N}(s_{j} + t_{j}) = 2bm, m is a positive integer, α = (α_{1}, …, α_{n}) is a multi-

index; |α| = α_{1} + α_{2} + ⋯ + α_{n}; D_{x}^{α} = D_{x_{1}}^{α_{1}} · D_{x_{2}}^{α_{2}} ⋯ D_{x_{n}}^{α_{n}}, D_{x_{j}} = −i\frac{∂}{∂x_{j}} (i is the imaginary unit).

The matrix L_{0}(x, t, ξ, σ) with elements

∑_{|α|+2bβ≤s_{k}+t_{j}} a_{kj}^{αβ}(x, t)ξ^{α}σ^{β} (k, j = 1, 2, …, N),

where ξ ∈ ℝ^{n}, σ is a complex-valued number, ξ^{α} = ξ_{1}^{α_{1}} ⋯ ξ_{n}^{α_{n}}, will be called the principal part of the symbol of system (1). It is assumed that system (1) is uniformly parabolic in the domain ω.

Following [6]–[9], we shall say that system (1) is parabolic in the domain ω if there exists a positive constant λ_{0} called a parabolicity constant such that for any (x, t) ∈ ω and ξ ∈ ℝ^{n} the roots σ_{1}, …, σ_{m} of the polynomial P (x, t, ξ, σ) = det L_{0}(x, t, ξ, σ) with respect to σ satisfy the inequality

Re σ_{s}(x, t, ξ) ≤ −λ_{0}|ξ|^{2b} (s = 1, 2, …, m),  (2)

where |ξ|^{2} = ∑_{j=1}^{n} ξ_{i}^{2}.

Denote by L̂_{0}(x, t, ξ, σ) the matrix of algebraic complements to the elements of the matrix L_{0}(x, t, ξ, σ). Then L̂_{0} = p · L_{0}^{−1}.

On the surface γ, we give general boundary conditions of the form

∑_{j=1}^{N} ∑_{|α|+2bβ≤q_{ν}+t_{j}} b_{ν_{j}}^{αβ}(x, t)D_{x}^{α} \frac{∂^{β}}{∂t^{β}} u_{j}(x, t)|_{γ} = 0 (ν = 1, 2, …, bm), (3)

where q_{ν} are some integers.

Let (x̂, t̂) ∈ γ. Consider the matrix B_{0}(x̂, t̂, ξ, σ) with

∑_{|α|+2bβ=q_{ν}+t_{j}} b_{ν_{j}}^{αβ}(x̂, t̂)ξ^{α}σ^{β}.

Let ν_{0} = (ν_{1}, …, ν_{n}) be the unit vector of the external normal to the surface s_{t} = γ ∩{x, t : t = t̂} at the point (x̂, t̂), and η(x̂, t̂) be any tangential vector to s_{t}at the same point. It follows from the parabolicity condition (see [6] and [10]) that the polynomial P (x̂, t̂, η(x̂, t̂) + τν_{0}(x̂, t̂), σ) with respect to τ has bm roots, τ_{+}^{s}(x̂, t̂, η, σ), (s = 1, 2, …, bm), with positive imaginary parts if there exists a constant λ_{1} such that 0 < λ_{1} < λ_{0} and the inequalities

Re σ ≥ −λ_{1}|η(x̂, t̂)|^{2b}, |σ|^{2} + |η(x̂, t̂)|^{4b} > 0

are fulfilled.

We set M^{+}(x̂, t̂, η, σ, τ) = ∏_{s=1}^{bm}(τ − τ_{+}^{s}(x̂, t̂, η, σ)).

As to the boundary conditions (3) and system (1), it is assumed that the conditions for being complementary are fulfilled for any point (x̂, t̂) ∈ γ. The condition for being complementary for system (1) and the boundary conditions (3) is fulfilled at the point (x̂, t̂) if the rows of the matrix

A(x̂, t̂, η(x̂, t̂) + τν_{0}(x̂, t̂), σ) ≡ B_{0}(x̂, t̂, η(x̂, t̂) + τν_{0}(x̂, t̂), σ) ×
×L̂_{0}(x̂, t̂, η(x̂, t̂) + τν_{0}(x̂, t̂), σ)

are linearly independent modulo the polynomial M^{+}(x̂, t̂, η, σ, τ) with respect to τ provided that Re σ ≥ −λ_{1}|η(x̂, t̂)|^{2b}, |σ|^{2} + |η(x̂, t̂)|^{4b} > 0.

We set

( ∑_{s=0}^{bm−1} q_{hj}^{s}(x̂, t̂, η, σ)τ^{s}) ≡ A(x̂, t̂, η(x̂, t̂) + τν_{0}(x̂, t̂), σ) ×

×(mod M^{+}(x̂, t̂, η, σ, τ)).

Consider the matrix Q(x̂, t̂, η, σ) with elements q_{b_{j}}^{s}(x̂, t̂, η, σ), which has bm rows and bmN columns (h = 1, 2, …, bm; s = 0, 1, …, bm − 1; j = 1, 2, …, N).

Let Δ_{k}(x̂, t̂, η, σ), (k = 1, 2, …, N_{1}) be the minors of order bm of the matrix Q and let Δ(x̂, t̂, η, σ) = max_{k} |Δ_{k}(x̂, t̂, η, σ)|.

According to [6] the condition for being complementary is fulfilled uniformly on the surface γ if

Δ_{γ} ≡ inf_{(_{x},_{t})∈γ} Δ(x̂, t̂, η(x̂, t̂), σ) > 0,  (4)
Re σ ≥ −λ|η(x̂, t̂)|^{2b}, |σ|^{2} + |η(x̂, t̂)|^{4b} = 1.

On ω_{0} = \overline{ω} ∩ \{x, t : t = 0\} we give initial conditions of the form

∑_{j=1}^{N} ∑_{|α|+2bβ≤r_{h}+t_{j}} c_{h_{j}}^{αβ}(x, t)D_{x}^{α} \frac{∂^{β}}{∂t^{β}} u_{j}(x, t)|_{ω_{0}} = 0 (h = 1, 2, …, m), (5)

where r_{h} are some negative integer numbers and the coefficients c_{h_{j}}^{αβ}(x, t) ≡ 0, when r_{h} + t_{j} < 0. Consider the matrix C_{0}(x, t, ξ, σ) with elements

∑_{|α|+2bβ=r_{h}+t_{j}} c_{h_{j}}^{αβ}(x, t)ξ^{α}σ^{β}.

For system (1) and the boundary conditions (5) the condition for being complementary is fulfilled at the point (x̂, t̂) ∈ \overline{ω} if the rows of the matrix H(x̂, t̂, σ)=C_{0}(x̂, t̂, 0, σ) · L̃_{0}(x̂, t̂, 0, σ) are linearly independent modulo σ^{m}.

Let ( ∑_{s=0}^{m−1} d_{h_{j}}^{s}(x̂, t̂)σ^{s}) ≡ H(x̂, t̂, σ)(mod σ^{m}).

Consider the matrix H(x̂, t̂) with elements d_{h_{j}}^{s}(x̂, t̂), which has m rows and mN columns (s = 0, 1, …, m − 1; j = 1, 2, …, N). Let E_{k}(x̂, t̂), (k = 1, 2, …, L_{1}) be the minors of order m of the matrix H(x̂, t̂) and let

Δ(x̂, t̂) = max_{k} |E_{k}(x̂, t̂)|.

According to [6] the condition for being complementary is fulfilled uniformly in \overline{ω} if

Δ_{0} ≡ inf_{\overline{ω}} Δ(x̂, t̂) > 0.  (6)

We introduce a space of functions, where a solution of problem (1), (3),

(5) will be considered.

Let G ⊂ ℝ_{x}^{n} × (0, T) be a finite domain. Denote by ⟨u⟩_{p,0}^{G} the norm of

the function u in the space L_{p}(G), ⟨u⟩_{p,0}^{G} = ( ∫_{G} |u|^{p} dx dt)^{1/p}.

We define the norm ∥u∥_{p,2bs}^{G} = ∑_{|α|+2bβ≤2bs}⟨D_{x}^{α} \frac{∂^{β}}{∂t^{β}} u⟩_{p,0}^{G}, where s is some

positive integer.

Denote by W_{p}^{2bs,s}(G) the space of functions obtained by completing with respect to the norm ∥u∥_{p,2bs}^{G} the set of smooth functions in \overline{G}.

As to the surface γ, it is assumed that it satisfies the conditions of uniform local unbending. By [9] this means the following:

In the space ℝ_{x,t}^{n+1} we consider the sets

H(x̂, t̂; ρ) = {x, t : |x_{j} − x̂_{j}| < ρ, j = 1, 2, …, n; −p^{2b} < t − t̂< ρ^{2b}};

H^{−}(x̂, t̂; ρ) = {x, t : |x_{j} − x̂_{j}| < ρ, j = 1, 2, …, n; −p^{2b} < t − t̂≤ 0};

H^{+}(x̂, t̂; ρ) = {x, t : |x_{j} − x̂_{j}| < ρ, j = 1, 2, …, n; 0 ≤ t − t̂< ρ^{2b}};

H^{+}(x̂, t̂; ρ_{1}, ρ_{2}) = {x, t : |x_{j} − x̂_{j}| < ρ_{1}, j = 1, 2, …, n; 0 ≤ t − t̂< ρ_{2}^{2b}}.

Analogous sets in the space ℝ_{y,τ}^{n+1} are denoted by s(ŷ, τ̂, ρ), s^{−}(ŷ, τ̂, ρ), s^{+}(ŷ, τ̂, ρ), and s^{+}(ŷ, τ̂, ρ_{1}, ρ_{2}), respectively.

The surface γ will be said to satisfy the condition of uniform local unbending with the constants d(ω), M(ω) and γ ∈ C^{s^{∗}+t^{∗},b}, where t^{∗} = max{t_{j}} and s^{∗} > q = max(0, q_{1}, …, q_{bm}), if the following conditions are fulfilled:

1. For any point (x̂, 0) ∈ ∂ω_{0} there exists a neighborhood O_{_{x},0} such that the set \overline{ω} ∩O_{_{x},0} is homeomorphic under some nondegenerate transformation of the coordinates Ψ_{_{x},0} = {y = f(x, t), τ = t : f(x̂, 0) = x̂} to the set s^{+}(0, 0; κ_{1}) ∩ \{y_{n} ≥ 0\}, 0 < κ ≤ 1, and the set \overline{γ} ∩ O_{_{x},0} is homeomorphic to s^{+}(0, 0; κ_{1}) ∩ {y_{n} = 0}. It is assumed that the number κ_{1} does not depend on the point (x̂, 0) ∈ ∂ω_{0}. Let

M_{0} = ∪_{(_{x},0)∈∂ω_{0}}Ψ_{_{x},0}^{−1}(s^{+}(0, 0; \frac{κ_{1}}{2} ) ∩ \{y_{n} ≥ 0\}).

2. For any point (x̂, T) ∈ ∂ω_{T}, where ω_{T} = \overline{ω} ∩ \{t = T \}, there exists a neighborhood O_{_{x},T} such that the set \overline{ω} ∩ O_{_{x},T} is homeomorphic under some nondegenerate transformation of the coordinates Ψ_{_{x},T} = {y = f(x, t), τ = t − T; f(x̂, T) = 0} to the sets s^{−}(0, 0; κ_{2}) ∩ {y_{n} ≥ 0}, and the set γ ∩ O_{_{x},T} is homeomorphic to s^{−}(0, 0; κ_{2}) ∩ {y_{n} = 0}, 0 < κ_{2} ≤ 1. It is assumed that the number κ_{2} does not depend on the point (x̂, T) ∈ ∂ω_{T}. Let

M_{T} = ∪_{(_{x},0)∈∂ω_{T}} Ψ_{_{x},T}^{−1} (s^{−}(0, 0; \frac{κ_{2}}{2} ) ∩ \{y_{n} ≥ 0\}).

3. The transformations Ψ_{_{x},0}, Ψ_{_{x},T}, Ψ_{_{x},0}^{−1}, Ψ_{_{x},T}^{−1} are given by the functions

whose norms in the space C^{s^{∗}+t^{∗},t} are bounded by the variable K_{1}(γ) ≥ 1. 4. For any point (x̂, t̂) ∈ γ_{1}, where

γ_{1} = γ ∩ \{x, t : \frac{1}{2}(\frac{κ_{1}}{2} )^{2b}(k_{1}(γ)(n + 1))^{−1} ≤ t ≤

≤ T − \frac{1}{2}(\frac{κ_{2}}{2} )^{2b}(k_{1}(γ)(n + 1))^{−1}\}

there exists a neighborhood O_{_{x},_{t}} such that the set O_{_{x},_{t}}∩\overline{ω} is homeomorphic under some nondegenerate transformation of the coordinates Ψ_{_{x},_{t}} = {y = f(x, t), τ = t − t̂; f(x̂, t̂) = 0} to the set s(0, 0; κ_{3}) ∩ {y_{n} ≥ 0}, 0 < κ_{3} ≤ 1, and the set O_{_{x},_{t}}∩γ is homeomorphic to s(0, 0; κ_{3})∩{y_{n} = 0}. Assume that κ_{3} does not depend on the point (x̂, t̂) ∈ γ_{1}. Let

M_{γ_{1}} = ∪_{(_{x},wht)∈γ_{1}}Ψ_{_{x},_{t}}^{−1}(s(0, 0; \frac{κ_{3}}{2} ) ∩ \{y_{n} ≥ 0\}).

5. The transformations ψ_{_{x},_{t}} and ψ_{_{x},_{t}}^{−1} are given by the functions whose

norms in the space C^{s^{∗}+t^{∗},t} are bounded by the variable K_{2}(γ) ≥ 1. Let K(γ) = max{K_{1}(γ), K_{2}(γ)}. Since the transformations ψ_{_{x},0}^{−1}, ψ_{_{x},T}^{−1}

and ψ_{_{x},_{t}}^{−1} shorten the distance between the points (n+1)K(γ) times at most, the distance from any point of the set ω_{1} = ω\\{M_{0} ∪ M_{T} ∪ M_{γ_{1}}\} to \overline{γ} is at least

d_{1}(γ) = min \{\frac{1}{2}(\frac{κ_{1}}{2} )^{2b}(K(j)(n + 1))^{−1}, \frac{1}{2}(\frac{κ_{2}}{2} )^{2b}(K(j)(n + 1))^{−1},

(\frac{κ_{3}}{2} )^{2b}(K(j)(n + 1))^{−1}\}.

We set d(ω) = d_{1}(γ)(n + 1)^{− \frac{1}{2}} , M(ω) = K(γ)(n + 1). It is easy to check that d(ω) < 1, and for any point (x_{0}, t_{0}) ∈ ω_{1} the parallelepiped H(x_{0}, t_{0}; ρ) ∩ {0 < t < T }) belongs to ω when ρ < d(ω).

The following statement is proved in [9].

Lemma 1. Let system (1) be uniformly parabolic in ω with the parabolicity constant λ_{0}. Let system (1), and the boundary and initial conditions (3) and (5) satisfy the conditions for being complementary to (4) and (6), respectively. It is assumed that the surface γ satisfies the condition of a uniform local unbending with the constants d(ω), M(ω), and γ ∈ C^{s^{∗}+t^{∗},b}, where t^{∗} = max(t_{1}, …, t_{N}), s^{∗} > q_{∗} = max(0, q_{b1}, …, q_{bn}). Let the integer numbers t_{j}, s_{k}, q_{ν}, and r_{h} be divisible by 2b, and for the coefficients of problem (1), (3), (5) the condition

∥a_{kj}^{αβ}; C^{s^{∗}−s_{k},b}(ω)∥ + ∥b_{νj}^{α′β′}; C^{s^{∗}−q_{ν},b}(γ)∥ +

+ sup_{τ∈[0,T]} ∥c_{hj}^{α′′β′′}; C^{s^{∗}−r_{h}}(\overline{ω} ∩ \{t = τ\})∥ ≤ M,

be fulfilled, where M = const > 0, k, j = 1, 2, …, N, ν = 1, 2, …, bm, h = 1, 2, …, m, |α|+2bβ ≤ s_{k}+t_{j}, |α′|+2bβ′ ≤ q_{ν} +t_{j}, |α′′|+2bβ′′ ≤ r_{h}+t_{j}.

Let l > 0, 0 < τ ≤ T, 1 ≤ p < ∞, and 0 < ρ ≤ min(1, d(ω), τ^{1/2b}). Then for any solution u of the homogeneous problem (1), (3), (5) such that

u_{j} ∈ W_{p}^{s^{∗}+t_{j},(s^{∗}+t_{j})/2b}(ω(l + M(ω)ρ; 0, τ)) (j = 1, 2, …, N),

we have an estimate

∑_{j=1}^{N} ρ^{s^{∗}+t^{∗}} ∥u_{j}; W_{p}^{s^{∗}+t_{j},(s^{∗}+t_{j})/2b}(ω(l; 0, τ))∥ ≤

≤ C_{0} ∑_{j=1}^{N} ρ^{s^{∗}−t_{j}} ∥u_{j}; L_{p}(ω(l + M(ω)ρ; 0, τ))∥,  (7)

where ω(l; 0, τ) = ω∩{x, t : |x_{j}| < l, j = 1, 2, …, n; 0 < t < τ}; the constant C_{0} depends on the set of constants {K} = {λ_{0}, n, s^{∗}, m, b, t_{1}, …, t_{N}, s_{1}, …, s_{N}, M, M(ω), d(ω)}.

Let us introduce an additional independent variable x_{0} and assume that the domain Ω = ω ×ℝ_{x_{0}}^{1}. In the domain Ω we consider an additional system of the form

∑_{j=1}^{N} ∑_{|α|+2bβ≤s_{k}+t_{j}} a_{kj}^{αβ}(x, t)D_{x}^{α}(\frac{∂^{β}}{∂t^{β}} + p_{1}D_{x_{0}}^{2b})ν_{j}(x, x_{0}, t) = 0  (8)
(k = 1, 2, …, N)

with the boundary conditions on Γ = γ × ℝ_{x_{0}}^{1}

∑_{j=1}^{N} ∑_{|α|+2bβ≤q_{ν}+t_{j}} b_{νj}^{αβ}(x, t)D_{x}^{α}(\frac{∂^{β}}{∂t^{β}} + p_{1}D_{x_{0}}^{2b})ν_{j}(x, x_{0}, t) = 0  (9)
(ν = 1, 2, …, bm),

and the initial conditions on Ω_{0} = ω × ℝ_{x_{0}}^{1}

∑_{j=1}^{N} ∑_{|α|+2bβ≤r_{h}+t_{j}} c_{hj}^{αβ}(x, t)D_{x}^{α}(\frac{∂^{β}}{∂t^{β}} + p_{1}D_{x_{0}}^{2b})ν_{j}(x, x_{0}, t) = 0 (10)
(h = 1, 2, …, m),

where p_{1} is a positive integer.

The following statement is also proved in [9].

Lemma 2.

(a) System (8) is uniformly parabolic in the domain Ω with the parabolicity constant λ(p_{1}); λ(p_{1}) → λ_{0} for p_{1} → ∞, where λ_{0} is the parabolicity constant of system (1).

(b) For system (8) and the initial conditions (10), the conditions for being complementary are fulfilled uniformly in Ω with the constant Δ_{0} defined by condition (6).

(c) There are positive constants A_{0} and p_{0} such that, for p_{1} ≥ p_{0}, system (8) and the boundary conditions (9) satisfy the conditions for being complementary uniformly on Γ with the constant Δ_{Γ} ≥ A_{0}Δ_{γ}, where Δ_{γ} is defined by condition (4).

(d) The coefficients of problem (8)–(10) satisfy the conditions of Lemma 1 with the constant M(p_{1}) > 0.

(e) The function v(x, x_{0}, t) = exp{iμx_{0} − p_{1}μ^{2b}t}u(x, t) is a solution of problem (8)–(10) for any real parameter μ if only u is a solution of problem (1), (3), (5).

Let u be a solution of problem (1), (3), (5). Consider two additional functions w(x, t) = exp{−p_{1}μ^{2b}t}u(x, t) and v(x, x_{0}, t) = exp{iμx_{0}}w(x, t).

By Lemma 2 the function v is a solution of problem (8)–(10) and thus this problem satisfies the conditions of Lemma 1. Then for v the inequality

∑_{j=1}^{N} ρ^{s^{∗}+t^{∗}} ∥v_{j}; W^{s^{∗}+t_{j},(s^{∗}+t_{j})/2b}(Ω(l; 0, τ))∥ ≤

≤ C_{0} ∑_{j=1}^{N} ρ^{s^{∗}−t_{j}} ∥v_{j}; L_{p}(Ω(l + M(ω)ρ; 0, τ))∥  (11)

holds, where Ω(l; 0, τ) = ω(l; 0, τ) × {|x_{0}| < l} and the constant C_{0} depends on the set of constant {K_{p_{1}}} = {λ(p_{1}), n, s^{∗}, t_{1}, …, t_{N}, s_{1}, …, s_{N}, b, m, M(p_{1}), s(ω), M(ω)}.

Lemma 3. For the functions w and v, for any l > 0 and 0 < τ ≤ T the estimate

∥v_{j}; L_{p}(Ω(l; 0, τ))∥ ≤ C_{1}^{j}∥w_{j}; L_{p}(ω(l; 0, τ))∥  (12)

holds, where the constant C_{1}^{j} depends on l and p.

Proof. It is easy to see that

∥v_{j}; L_{p}(Ω(l; 0, τ))∥ = ∥ exp{iμx_{0}}w_{j}; L_{p}(Ω(l; 0, τ))∥ =

= ( ∫_{Ω(l;0,τ)} |w_{j}|^{p}dx dx_{0} dt)^{1/p} = e^{1/p}( ∫_{ω(l;0,τ)} |w_{j}|^{p}dx dt)^{1/p}.

Hence follows inequality (12).

Lemma 4. For the functions v and w, for any l > 0, μ > 0, 0 < τ ≤ T the inequality

C_{2}^{j}(μ, l)∥w_{j}; L_{p}(ω(l; 0, τ))∥ ≤ ∥v_{j}; W_{p}^{s^{∗}+t_{j},(s^{∗}+t_{j})/2b}(Ω(l; 0, τ))∥ (13)

holds, where the constant C_{2}^{j} depends only on l, m, p.

Proof. As in proving Lemma 3, it is easy to see that

∥v_{j}; W_{p}^{s^{∗}+t_{j},(s^{∗}+t_{j})/2b}(Ω(l; 0, τ))∥ = ∑_{|α|+2bβ≤s^{∗}+t_{j}}⟨D_{x,x_{0}}^{α} \frac{∂^{β}}{∂t^{β}} v_{j}⟩_{p,0}^{Ω(l;0,τ)} ≥

≥ ⟨D_{x_{0}}^{s^{∗}+t_{j}}v_{j}⟩_{p,0}^{Ω(l;0,r)} = ( ∫_{Ω(l;0,τ)} μ^{(s^{∗}+t_{j})p}|w_{j}|^{p}dx_{0} dx dt)^{1/p} =

= μ^{s^{∗}+t_{j}}e^{1/p}∥w_{j}; L_{p}(ω(l; 0, τ))∥.

Hence follows inequality (13) for C_{2}^{j} = μ^{s^{∗}+t_{j}}l^{\frac{1}{p}}.

Lemma 5. For the function w, for any l > 0, 0 < τ ≤ T, and 0 < ρ ≤ min(1, d(ω), τ^{1/2b}) the inequality

∑_{j=1}^{N} C_{3}^{j}(μ, ρ, l)ρ^{t^{∗}−t_{j}} ∥w_{j}; L_{p}(ω(l; 0, τ))∥ ≤

≤ ∑_{j=1}^{N} ρ^{t^{∗}−t_{j}} ∥w_{j}; L_{p}(ω(l + M(ω)ρ; 0, τ))∥  (14)

holds, where the constant C_{3}^{j} depends only on μ, ρ, l, p.

Proof. With (12) and (13) taken into account, inequality (11) implies

∑_{j=1}^{N} ρ^{s^{∗}+t^{∗}}C_{2}^{j}(μ, l)∥w_{j}; L_{p}(ω(l; 0, τ))∥ ≤

≤ C_{0} ∑_{j=1}^{N} ρ^{t^{∗}−t_{j}}C_{1}^{j}(l + M(ω)ρ)∥w_{j}; L_{p}(ω(l + M(ω)ρ; 0, τ))∥.

Hence follows inequality (14) for C_{3}^{j} = e^{\frac{1}{p}} (l + M(ω)ρ)^{−1/p} \frac{1}{C_{0}} (μρ)^{s^{∗}+t_{j}}.

Lemma 6. Let u be a solution of problem (1), (3), (5). Then for the function w, for any m_{1} ≥ m_{0} = const > 0 and 0 < τ ≤ T the inequality

∑_{j=1}^{N} ∥w_{j}; L_{p}(ω(2^{m_{1}}; 0, τ))∥≤e^{− \frac{k(m_{1})}{2}} ∑_{j=1}^{N} ∥w_{j}; L_{p}(ω(2^{m_{1}+1}, 2^{m_{1}}; 0, τ))∥ (15)

holds, where k(m_{1}) = [λ2^{\frac{2b}{2b−1} m_{1}}] and λ is a positive number, while ω(l_{2}, l_{1}; 0, τ) = ω(l_{2}; 0, τ)\ω(l_{1}; 0, τ) for l_{2} > l_{1}.

Proof. Assume l_{1} = 2^{m_{1}} and l_{2} = 2l_{1}. Let ρ(m_{1}) = 2^{m_{1}}/(M(ω)k(m_{1})).

Let m_{0} be a sufficiently large number such that for m_{1} ≥ m_{0} the inequality

2^{m_{1}}/(M(ω)K(m_{1})) ≤ min(1, d(ω), τ^{\frac{1}{2b}})  (16)

is fulfilled.

Let l_{1}(n_{1}) = l_{1} + n_{1}M(ω)ρ and l_{1}(0) = l_{1}. Then inequality (14) implies

∑_{j=1}^{N} C_{3}^{j}ρ^{t^{∗}−t_{j}} ∥w_{j}; L_{p}(ω(l_{1} + n_{1}M(ω)ρ; 0, τ))∥ ≤

≤ ∑_{j=1}^{N} ρ^{t^{∗}−t_{j}} ∥w_{j}; L_{p}(ω(l_{1} + (n + 1)M(ω)ρ; 0, τ))∥.

It is easy to see that

C_{3}^{j} = (\frac{2^{m_{1}} + n_{1}M(ω)ρ}{2^{m_{1}} + (n_{1} + 1)M(ω)ρ})^{\frac{1}{p}} \frac{1}{C_{0}} (μρ)^{s^{∗}+t_{j}} ≥ 2^{− \frac{1}{p}} C_{0}^{−1}(μρ(m_{1}))^{s^{∗}+t_{j}}.

Let μ = λb_{0}2^{\frac{m_{1}}{2b−1}} . Since ρ(m_{1})≥2^{m_{1}}(M(ω)λ)^{−1}2^{− \frac{2bm_{1}}{2b−1}} =(M(ω)λ2^{\frac{m_{1}}{2b−1}} )^{−1}, we have C_{3}^{j} ≥ 2^{− \frac{1}{p}} C_{0}^{−1}b_{0}^{s^{∗}+t_{j}}. It is easy to check that if b_{0} = (2^{\frac{p+1}{p}} C_{0}e)^{\frac{1}{s^{∗}}}, then C_{3}^{j} ≥ 2e for any j = 1, 2, …, N. In that case we obtain

∑_{j=1}^{N} 2eρ^{t^{∗}−t_{j}} ∥w_{j}; L_{p}(ω(l_{1}(n_{1}); 0, τ))∥ ≤ ∑_{j=1}^{N} ρ^{t^{∗}−t_{j}} ∥w_{j}; L_{p}(ω(l_{2}(n_{1}); 0, τ))∥,

where l_{2}(n_{1}) = l_{1}(n_{1} + 1). Hence we find

∑_{j=1}^{N}(2e − 1)ρ^{t^{∗}−t_{j}} ∥w_{j}; L_{p}(ω(l_{1}(n_{1}); 0, τ))∥ ≤

≤ ∑_{j=1}^{N} ρ^{t^{∗}−t_{j}} ∥w_{j}; L_{p}(ω(l_{2}(n_{1}), l_{1}(n_{1}); 0, τ))∥.

But since 2e − 1 > 0, we have

∑_{j=1}^{N} ρ^{t^{∗}−t_{j}} ∥w_{j}; L_{p}(ω(l_{1}(n_{1}); 0, τ))∥ ≤

≤ e^{−1} ∑_{j=1}^{N} ρ^{t^{∗}−t_{j}} ∥w_{j}; L_{p}(ω(l_{2}(n_{1}), l_{1}(n_{1}); 0, τ))∥.  (17)

If we now apply inequality (17) corresponding to n_{1} = ν + 1 to estimate the right-hand side of this inequality corresponding to n_{1} = ν and assume successively that ν = 0, 1, 2, …, (k − 1) and also take into account the fact that the domain ω(l_{2}(k(m_{1}) − 1), l_{1}(k(m_{1}) − 1); 0, τ) is contained in the domain ω(l_{2}, l_{1}; 0, τ), then, since ρ ≤ 1, we obtain

∑_{j=1}^{N}ρ^{t^{∗}−t)j}∥w_{j}; L_{p}(ω(2^{m_{1}}; 0, τ))∥≤e^{−k(m_{1})}∑_{j=1}^{N}∥w_{j}; L_{p}(ω(2^{m_{1}+1}, 2^{m_{1}}; 0, τ))∥.

Since t^{∗} − t_{j} ≥ 0 for any j, we have

(ρ(m_{1}))^{t^{∗}−t_{j}} ≥ (2^{m_{1}}/(M(ω)k(m_{1})))^{t^{∗}−t_{j}} ≥ (M(ω)k(m_{1}))^{t_{j}−t^{∗}}.

Since k(m_{1}) → ∞ as m_{1} → ∞, the estimate (M(ω)k(m_{1}))^{t^{∗}−t_{j}} ≤ exp(k(m_{1})/2) holds for m_{1} ≥ m_{0}, where m_{0} is a sufficiently large number.

Thus for m_{1} ≥ m_{0} the function w satisfies the inequality

∑_{j=1}^{N}∥w_{j}; L_{p}(ω(2^{m_{1}}; 0, τ))∥≤e^{− \frac{k(m_{1})}{2}}\} ∑_{j=1}^{N}∥w_{j}; L_{p}(ω(2^{m_{1}+1}, 2^{m_{1}}; 0, τ))∥.

Theorem. Let the domain ω be such that for a solution u of problem (1), (3), (5) in the domain ω the equality

lim_{R→∞} exp\{−σR^{\frac{2b}{2b−1}} \} · ∥u; L_{p}(ω(2R, R; 0, T))∥ = 0  (18)

holds, where 1 ≤ ρ < ∞ and σ is a positive constant. Then u ≡ 0 in ω.

Proof. Since w(x, t) = exp{−p_{1}μ^{2b}t}u(x, t), inequality (15) implies

∑_{j=1}^{N} ∥u_{j}; L_{p}(ω(2^{m_{1}}; 0, τ))∥ ≤ exp \{ − \frac{1}{2}k(m_{1}) + p_{1}μ^{2b}(m_{1})τ\} ×

× ∑_{j=1}^{N} ∥u_{j}; L_{p}(ω(2^{m_{1}+1}, 2^{m_{1}}; 0, τ))∥,

where k(m_{1}) = [2^{\frac{2bm_{1}}{2b−1}} λ], μ(m_{1}) = λb_{0}2^{\frac{m_{1}}{2b−1}} .

Let λ = 4σ and τ_{0} = (4p_{1}λ^{2b−1}b_{0}^{2b})^{−1}. Then for any τ ≤ τ_{0} we have

∑_{j=1}^{N} ∥u_{j}; L_{p}(ω(2^{m_{1}}; 0, τ))∥ ≤ exp \{ − σ2^{\frac{2b}{2b−1} m_{1}}\} ×

× ∑_{j=1}^{N} ∥u_{j}; L_{p}(ω(2^{m_{2}+1}, 2^{m_{1}}; 0, τ))∥.

Passing in the latter inequality to the limit as m_{1} → ∞ and taking into account condition (18), we find that u ≡ 0 in the domain ω ∩ {x, t : 0 ≤ t ≤ τ_{0}}. Next we find that u ≡ 0 in the domains ω ∩ {x, t : τ_{0} ≤ t ≤ 2τ_{0}}, ω ∩ {x, t : 2τ_{0} ≤ t ≤ 3τ_{0}}, . . .. Therefore u(x, t) ≡ 0 in ω.

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(Received 18.04.1996)

Author’s address:

I. Javakhishvili Tbilisi State University

Faculty of Mechanics and Mathematics

2, University St., Tbilisi 380043

Georgia