Electronic Journal of Qualitative Theory of Differential Equations Proc. 8th Coll. QTDE, 2008, No. 13 1-10; http://www.math.u-szeged.hu/ejqtde/

ON THE GLOBAL EXISTENCE OF MILD SOLUTIONS OF NONLINEAR DELAY SYSTEMS ASSOCIATED WITH CONTINUOUS AND ANALYTIC SEMIGROUPS

Milan Medveď

Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University, Bratislava, Slovakia, e-mail: medved@fmph.uniba.sk

Abstract.

In this paper we prove sufficient conditions for the existence of global solutions of nonlinear functional-differential evolution equations whose linear parts are infinitesimal generators of strongly continuous and analytic semigroups. We apply the obtained results to a diffusion problem.

2000 MSC: Primary: 47J35, 34K12, Secondary: 34K90, 35K20, 35K60

1. Introduction

Integral inequalities play an important role in the study of asymptotic behavior of integral and differential equations. In the paper [11] a singular version of the Bihari nonlinear integral inequality (see [1]) is proved. Some modifications of this inequality are applied in the study of the existence of global solutions of semilinear evolution equations and their stability properties in the papers [7, 11, 12, 13, 14] These results are closed to the results obtained in the paper [4] by using a comparison method. In the paper [15] the following delay system is studied:

(1)  ẋ(t) = Ax(t) + f(t, x(t), x_{t}), t ≧ 0,
x(t) = Φ(t), t ∈ ⟨−r, 0⟩,

where x(t) ∈ X (X is a Banach space), x_{t} ∈ C := C(⟨−r, 0⟩, X), x_{t}(Θ) = x(t + Θ), r ≦ Θ ≦ 0, Φ ∈ C, ||u|| = sup_{−r≦Θ≦0} ||u(Θ)||, u ∈ C.

Using the technique of integral inequalities sufficient conditions for the existence of global mild solutions of this problem are proved. This problem is studied also in the paper [19] by applying a result on linear integral inequality with singular kernel, where it is assumed that the map f(t, u, v) is linearly bounded in the variables u, v. Our conditions on the map f are more general. In the last section we apply our results to a diffusion problem which is a modification of the problem for differential equations of diffusion type without a delay, studied in the paper [8], to a similar problem for functional-differential equations.

Typeset by AMS-TEX

EJQTDE Proc. 8th Coll. QTDE, 2008 No. 13, p. 1

Key words and phrases. continuous and analytic semigroup, functional-differential equation, global solution, boundary value problem, parabolic equation.

This paper is in final form and no version of it is submitted for publication elsewhere.

2. On the existence of global solutions

Let {S(t)}_{t∈R^{+}}, S(t) ∈ L := L(X, X) be a strongly continuous semigroup (see e. g. [2, 10, 15, 16])

and

Ax := lim_{t→0^{+}} \frac{1}{t}[S(t)x − x], x ∈ D(A)

be generator of the semigroup We denote S(t) := e^{At}. By [16, Theorem 2. 2], or [2, p. 22] there exist constants M ≧ 1, α ≧ 0 such that

(2)  ||e^{At}||_{L} ≦ Me^{αt}, t ≧ 0.

By a mild solution of the initial value problem (1) on an interval ⟨−r, T), T > 0 we mean a continuous function x ∈ C(⟨−r, T), X) satisfying

x(t) = e^{At}Φ(0) + ∫_{0}^{t} e^{A(t−s)}f(s, x(s), x_{s})ds, 0 ≦ t < T,

x(t) = Φ(t), −r ≦ t ≦ 0,

We assume that the mapping f : R^{+} × X × C ↦ X satisfies the conditions:

(H1) There exist continuous, nonnegative functions F_{1}(t), F_{2}(t), t ≧ 0 and continuous, positive, nondecreasing functions ω_{1}, ω_{2} : R^{+} → R^{+} such that

||f(t, x, y)|| ≦ F_{1}(t)ω_{1}(||x||) + F_{2}(t)ω_{2}(||y||_{C})

for all t ∈ R^{+}, x ∈ X, y ∈ C, (H2)

∫_{0}^{∞} \frac{dσ}{ω_{1}(σ) + ω_{2}(σ)} = ∞,

In the paper [15] we have assumed the condition (H2) with ω_{1} = ω_{2}.

Definition. A solution x : ⟨0, T) → X of the equations (1) is called nonextendable, if either 0 < T < ∞and then lim sup_{t→T^{−}} ||x(t)|| = ∞, or T = ∞. In the second case the solution is called global.

We do not study the problem of the existence of solutions of the problem (1). In the papers [18, 19] the problem of the existence of maximal and noncontinuable (nonextendable) solutions for functional-differential equations with the right-hand sides of the form f(t, x_{t}) is studied.

The following theorem is proved in the paper [15].

THEOREM 1 ([15, Theorem 1]). Let A : D(A) → X be the infinitesimal generator of a strongly continuous semigroup {e^{At}}_{t≧0} and the condition (H1), (H2) with ω := ω_{1} = ω_{2}, be satisfied. Then any nonextendable solution x : ⟨0, T) → X of the initial value problem (1) is global.

Theorem 2. Let the assumptions of Theorem 1 be fulfilled, where the condition (H2) is with ω_{1} ≠ ω_{2}, i. e.

∫_{0}^{∞} \frac{dσ}{ω_{1}(σ) + ω_{2}(σ)} = ∞.

Then any nonextendable solution x : ⟨0, T) → X of the initial value problem (1) is global.

If we put F = max{F_{1}, F_{2}}, ω = ω_{1} +ω_{2} then we obtain the assertion of theorem 2 as a consequence of Theorem 1. The formulation of Theorem 2 has a reason because the variables y, z of the mapping f(t, y, z) belong to different spaces and the appearance of different functions ω_{1}, ω_{2} in the condition (H_{2}) is natural.

Before giving an example of different functions ω_{1}, ω_{2} satisfying the condition (H2), let us recall a useful result by A. Constantin [3] (see also [4, Lemma 1]).

Proposition. Let ω ∈ C(R_{+}, (0, ∞)) be nondecreasing and ∫_{0}^{∞} \frac{ds}{ω(s)} = ∞, then ∫_{0}^{∞} \frac{ds}{s+ω(s)} = ∞.

Example. Let ω_{1}(u) := u, ω_{2}(u) be a continuous, nondecreasing, positive function such that ∫_{0}^{∞} \frac{ds}{ω_{2}(s)} = ∞. Then the condition (H2) is satisfied.

Now let us consider the case when the operator A : D(A) ⊆ X → X is sectorial. Consider the initial value problem (1) in the form

(6)  ẋ(t) + Ax(t) = f(t, x(t), x_{t}), t ≧ 0,
x(t) = Φ(t), t ∈ ⟨−r, 0⟩, r > 0

in accordance with the books [6, 7] and some other books and papers dealing with the problem, where the linear part of the equation is a sectorial operator.The definition of the sectorial operator can be found i. e. in [7]. By [7, Theorem 1.3.4] if A is sectorial then −A is the infinitesimal generator of an analytic semigroup {e^{−A}}_{t≧0} (analytic means that t ↦ e^{−At}x is an analytic function on (0, ∞) for any x ∈ X.)

If A is sectorial on X and Re σ(A) > 0 and α ∈ ⟨0, 1) then one can define the operator

A^{−α} := \frac{1}{Γ(α)} ∫_{0}^{∞} t^{α−1}e^{−At}dt,

where Γ(α) is the Eulerian gamma function evaluated at α. The operator A^{−α} is bounded linear operator on X which is one-to-one and one can define A^{α} as the inverse of A^{−α} and we define A^{0} = I. The operator A^{α} is closed and densely defined.

If A is sectorial on X then there is an a ∈ R such that A_{1} = A + aI has Re σ(A_{1}) > 0 and thus the power A_{1}^{α} is defined. If we define X^{α} = D(A_{1}^{α}) and ||x||_{α} := ||A_{1}^{α}x||, x ∈ X^{α} then (X^{α}, || . ||_{α}) is a Banach space and there are constants M ≧ 1, γ > 0 such that the following holds:

(7)  ||e^{−At}x||_{α} ≦ Me^{−γt}||x||_{α},

(8)  ||e^{−At}x||_{α} ≦ Mt^{−α}e^{−γt}||x||

EJQTDE Proc. 8th Coll. QTDE, 2008 No. 13, p. 3 for all t > 0, x ∈ X^{α}, (see [6, Theorem 1.5.4], or [10, Proposition 2.1.1]).

By a mild solution of the initial value problem (6)on the interval ⟨−r, T) we mean a continuous function x : ⟨0, T) → X^{α} for which the function

f( . , x( .), x_{.}) : ⟨−r, T) → X, t ↦ f(t, x(t), x_{t})

is continuous, where

f : R^{+} × X^{α} × C_{α} → X

(C_{α} := C(⟨−r, 0), X^{α}) with the norm ||Φ||_{α} := sup_{−r≦Θ≦0}||Φ(Θ)||_{α}) and

x(t) = e^{−At}Φ(0) + ∫_{0}^{t} e^{−A(t−s)}f(s, x(s), x_{s})ds, t ∈ ⟨0, T),
x(t) = Φ(t), t ∈ ⟨−r, 0⟩.

We assume

(G1)

||f(t, x, y)|| ≦ F_{1}(t)ω_{1}(||x||_{α}) + F_{2}(t)ω_{2}(||y||_{C_{α}})

for all t ≧ 0, x ∈ X^{α}, y ∈ C_{α}, where F_{1}(t), F_{2}(t) and ω_{1}, ω_{2} are as above.

(G2)

∫_{0}^{∞} \frac{τ^{q−1}dτ}{ω_{1}(τ)^{q} + ω_{2}(τ)^{q}} = ∞,

where q = q(ϵ) = \frac{1}{β} + ϵ, ϵ > 0, β = 1 − α.

The definition of nonextendable and global solution is the same as for the initial value problem (1). In the paper [15] the following theorem is proved.

THEOREM 3 ([15, Theorem 3]). Let the conditions (G1) and (G2) with ω_{1} = ω_{2} be fulfilled and the inequalities (7), (8) hold. Then any nonextendable solution x : ⟨0, T) → X of the initial value problem (6) is global.

We shall prove the following theorem.

THEOREM 4. Let the conditions (G1) and (G2) with ω_{1} ≠ ω_{2}, be fulfilled, i. e.

∫_{0}^{∞} \frac{τ^{q−1}dτ}{ω_{1}(τ)^{q} + ω_{2}(τ)^{q}} = ∞

and the conditions (7), (8) hold. Then any nonextendable solution x : ⟨0, T) → X^{α} of the initial value problem (6) is global.

Proof. Let x : ⟨0, T) → X^{α} be a nonextendable solution with 0 < T < ∞. Then lim_{t→T^{−}} ||x(t)||_{α} = ∞.

||x(t)||_{α} ≦ ||e^{−At}Φ(0)||_{α} + ∫_{0}^{t} ||e^{A(t−s)}f(s, x(s), x_{s})||_{α}ds.

Applying the property of f and the inequalities (7), (8) we obtain

||x(t)||_{α} ≦ Me^{−γt}||Φ(0)||_{α} + M ∫_{0}^{t}(t − s)^{−α}e^{−γs}||f(s, x(s), x_{s})||ds ≦

EJQTDE Proc. 8th Coll. QTDE, 2008 No. 13, p. 4 ≦ Me^{−γt}||Φ(0)||_{α} + M ∫_{0}^{t}(t − s)^{−α}e^{−γs}[F_{1}(s)ω_{1}(||x(s)||_{α}) + F_{2}(s)ω_{2}(||x_{s}||_{C_{α}})]ds.

Now we shall apply a desingularization method developed in the paper [11] for integral inequalities with singular kernels.

Let β = 1 − α = \frac{1}{1+z} , z > 0, p = \frac{1+z+ϵ}{z+ϵ}, q = q(ϵ) = 1 + z + ϵ = \frac{1}{β} + ϵ, ϵ > 0 then \frac{1}{p} + \frac{1}{q} = 1 and using the Hölder inequality we obtain

∫_{0}^{t}(t − s)^{−α}F_{1}(s)ω_{1}(||x(s)||_{α})ds ≦

≦ (∫ (t − s)^{−αp}e^{ps})^{\frac{1}{p}} (∫_{0}^{t} F_{1}(s)e^{−qs}ω_{1}(||x(s)||_{α})^{q}ds)^{\frac{1}{q}} .

Analogously for the second integral. Since

∫_{0}^{t}(t − s)^{−αp}e^{ps}ds = e^{pt} ∫_{0}^{t} τ^{−αp}e^{−pτ} dτ < Qe^{pt},

where Q = \frac{Γ(1−αp)}{p^{1−αp}} (αp = \frac{z(1+z+ϵ)}{(1+z)(z+ϵ)}) ≠ 1). If we denote

a = Qe^{T} , C_{i}(t) = F_{i}(t)^{q},

we obtain

||x(t)||_{α} ≦ M||Φ(0)||_{α}+

+ (∫_{0}^{t} C_{1}(s)ω_{1}(||x(s)||_{α})^{q}ds)^{\frac{1}{q}} + (∫_{0}^{t} C_{2}(s)ω_{2}(||x_{s}||_{C_{α}})^{q})^{\frac{1}{q}} .

Applying the inequality (A + B + C)^{q} ≦ 3^{q−1}(A^{q} + B^{q} + C^{q}), A, B, C ≧ 0 we have

(*) ||x(t)||_{α}^{q} ≦ a + ∫_{0}^{t} D_{1}(s)ω_{1}(||x(s)||_{α})^{q}ds + ∫_{0}^{t} D_{2}(s)ω_{2}(||x_{s}||_{C_{α})})^{q}ds,

where a = 3^{q−1} (M||Φ(0)||_{α})^{q} , D_{i} = 3^{q−1}C_{i}, i = 1, 2. Now we can proceed as in the proof of [15, Theorem 3]. If we denote the right-hand side of the inequality (*) by g(t) then

||x(t)||_{α}^{q} ≦ g(t).

Without loss of generality we can choose M ≧ 1 so large that ||Φ(t)||_{C} ≦ a. Then we have

||x_{t}||_{C_{α}}^{q} = max{||Φ||_{C}^{q}, sup_{0≦τ≦t} ||x(τ)||^{q}} ≦ max{a, g(t)} = g(t).

Now one can proceed in the same way as in the proof of [15, Theorem 3] to obtain

lim sup_{t→T^{−}} Λ(||x(t)||_{α}^{q}) = lim sup_{t→T^{−}} ∫_{0}^{||x(t)||_{α}^{q}} \frac{dσ}{ω(σ^{\frac{1}{q}})} =

= ∫_{0}^{∞} \frac{dσ}{ω(σ^{\frac{1}{q}})} = q ∫_{0}^{∞} \frac{τ^{q−1}dτ}{ω(τ)} = ∞,

where ω = ω_{1}^{q} + ω_{2}^{q}. The contradiction.

Remark: If we put ω = ω_{1}^{q} +ω_{2}^{q}, F_{1} = D_{1}, F_{2} = D_{2} in the condition (G1)(see the proof of Theorem 4) we obtain Theorem 4 as a consequence of Theorem 3. However the proof of Theorem 4 says us why this function has the form ω = ω_{1}^{q} + ω_{2}^{q}, if the condition (G1) is formulated with different functions ω_{1}, ω_{2}. The case of the strongly continuous semigroup, considered above is more clear. We remark that the variables y, z of the mapping f(t, y, z) belong to different spaces.

Example:

Let

ω_{1}(u) = u, ω_{2}(u) = u^{\frac{q−1}{q}} (1 + ln(1 + u))^{\frac{1}{q}} .

Then

∫_{0}^{∞} \frac{σ^{q−1}dσ}{ω_{1}(σ)^{q} + ω_{2}(σ)^{q}} = ∫_{0}^{∞} \frac{dσ}{σ + 1 + ln(1 + σ)} = ∫_{1}^{∞} \frac{dτ}{τ + ln τ} = ∞

and thus the condition (G2) is satisfied.

4. Application: Boundary value problem for a delay system of parabolic equations

The following application of our results concerning differential equations, whose linear parts are represented by sectorial operators, is motivated by the paper [8], where some stability results for systems of differential equations studied below, however without delay, are proved. We also use some notations from this paper. In the paper [9] there are many nice examples for equations, whose linear parts are represented by continuous semigroups. It would be possible to apply our results from the first part of this paper to this type of equations but we concentrate to the following application only.

Consider the following delay system of parabolic equations with homogeneous Neumann boundary conditions:

\frac{∂u(t, x)}{∂t} = DΔu(t, x) + f(t, u(t, x), u_{t}), t ≧ 0, u ∈ R^{n}, x ∈ Ω ⊂ R^{N} a bounded set,

\frac{∂u}{∂η} = 0, x ∈ ∂Ω,

(9)  u(t, x) = Φ(t, x), −r ≦ t ≦ 0, x ∈ Ω,

\frac{∂Φ}{∂η} = 0, x ∈ ∂Ω,

where u_{t} ∈ C = C(⟨−r, 0⟩ × Ω, R^{n}), u_{t}(Θ, x) = u(t + Θ, x), N = 1, 2, or 3, D = diag{d_{1}, d_{2}, …, d_{n}} is a diagonal matrix, d_{i} > 0, i = 1, 2, …, n,

Let

X = Ł^{2}(Ω) := L^{2}(Ω, R), A : D(A) ⊂ X → X, Aϕ = −Δϕ,

EJQTDE Proc. 8th Coll. QTDE, 2008 No. 13, p. 6

D(A) = \{ϕ ∈ H^{2}(Ω, R) : \frac{∂ϕ}{∂η} = 0, on ∂Ω\},

H^{2} := H^{2}(Ω, R) := W^{2,2}(Ω, R) is the Sobolev space of g ∈ L^{2}(Ω, R) possessing distributional derivatives D^{j}g(x)of order ≦ 2 with the norm

||g||_{H^{2}} = ∑_{0≦|j|≦2} (∫_{Ω} ||D^{j}g(x)||^{2}dx)^{\frac{1}{2}} ,

j = (j_{1}, j_{2}, …, N)), |j| = j_{1} + j_{2} + ⋯ + j_{N}.

This system, however without delay, is studied in the paper [8] and the approach we are using in the following considerations also comes from this paper.

The operator A has eigenvaues

0 = λ_{1} < λ_{2} < ⋯ < λ_{k} → ∞

with the property that λ_{j} has finite multiplicity γ_{j} (which equals to the dimension of the corresponding eigenspace). Further, there is a complete set {Φ_{j,k}} of eigenvectors of A. This yields that if x ∈ D(A) then

Ax = ∑_{j=1}^{∞} λ_{j} ∑_{k=1}^{γ_{j}}⟨x, Φ_{j,k}⟩Φ_{j,k} = ∑_{j=1}^{∞} λ_{j}E_{j}x,

where ⟨ . , . ⟩ is the inner product in X and

E_{j}x = ∑_{k=1}^{γ_{j}}⟨x, Φ_{j,k}⟩Φ_{j,k}.

The family {E_{j}} is a family of complete orthonormal projections in X and

x = ∑_{j=1}^{∞} E_{j}x, x ∈ X.

The operator −A generates an analytic semigroup {e^{−At}}_{t≧0} defined by

e^{−At}x = E_{1}x + ∑_{j=2}^{∞} e^{−λ_{j}t}E_{j}x.

For a > 0, A_{1} = A + aI, 0 < α < 1 the fractional space

X^{α} = D(A_{1}^{α}) = {x ∈ X : ∑_{j=1}^{∞}(λ_{j} + a)^{2α}||E_{j}x||^{2} < ∞}

is defined and

A_{1}^{α}x = ∑_{j=1}^{∞}(λ_{j} + a)^{α}E_{j}x.

Let

Z = L^{2}(Ω, R^{n}) = X^{n} = X × X × ⋯ × X, C := C(Ω, R^{n}) = C(Ω)^{n}

with the usual norms.

Define

A_{D} : D(A_{D}) ⊂ Z → Z, A_{D}ψ = −DΔψ = DAψ,

where

D(A_{D}) = \{ϕ ∈ H^{2}(Ω, R^{n}) : \frac{∂ϕ}{∂η} = 0, on ∂Ω\},

−A_{D} is sectorial with the fractional power

Z^{α} = D(A_{D1}^{α}) = X^{α} × … X^{α} = [X^{α}]^{n}, A_{D1} = A_{D} + aI,

||z||_{α} = ||A_{D1}^{α}z||, z ∈ Z^{α},

A_{D1}^{α}z = ∑_{j=1}^{∞} D^{α}(λ_{j} + a)^{α}P_{j}z,

D^{α} = diag{d_{1}^{α}, d_{2}^{α}, …, d_{n}^{α}}, P_{j} = diag{E_{j}, E_{j}, …, E_{j}}, n × nmatrix.

The operator generates the analytic semigroup {e^{−A_{D}t}}_{t≧0},, where

e^{−A_{D}t}z = P_{1}z + ∑_{j=2}^{∞} e^{−λ_{j}Dt}P_{j}z, z ∈ Z.

The family {P_{j}} is a family of orthonormal projections in Z which is complete and therefore

z = ∑_{j=1}^{∞} ||P_{j}z, ||z|| = ∑_{j=1}^{∞} ||P_{j}z||^{2}, ||z||_{α} = ∑_{j=1}^{∞} ||P_{j}z||_{α}^{2},

For \frac{3}{4} < α < 1 the following inequalities holds:

||e^{−A_{D}t}z||_{α} ≦ M||z||_{α}, ≧ 0,
||e^{−A_{D}t}z||_{α} ≦ Mt^{−α}||z||, t > 0

([6, Theorem 1.6.1]) and the inclusions

Z^{α} ⊂ C(Ω, R^{n}), Z^{α} ⊂ L^{p}(Ω, R^{n}), p ≧ 2

are continuous (see [6]).

The abstract formulation of the problem:

ż + A_{D}y = g(t, z, z_{t}), t ≧ 0,

z(t) = Φ(t), −r ≦ t ≦ 0, z ∈ Z^{α},
z(x)(t) = z(t, x), z(x)_{t}(Θ) = y(t + Θ, x),

g : R^{+} × Z^{α} × C_{α} → Y, g(t, z, y) = f(t, z(x), y(x)_{t}), x ∈ Ω, y(x) ∈ C_{α}. EJQTDE Proc. 8th Coll. QTDE, 2008 No. 13, p. 8 From the continuous inclusion Z^{α} ⊂ C(Ω)^{n} it follows that there exists l > 1 such that

sup_{x∈Ω} ||z(x)||_{R^{n}} ≦ l||z||_{α}, z ∈ Z^{α}.

If we assume

f(t, z, y) = f_{1}(t, z) + f_{2}(t, y)

and

||f_{1}(t, z)|| ≦ F_{1}(t)ω_{1}(||z||), ||f_{2}(t, y)|| ≦ F_{2}(t)ω_{2}(||y||_{C}),

then

g_{1}(t, z) = f_{1}(t, z(x)), g_{2}(t, y) = f_{2}(t, y)

and

(11)  ||g_{1}(t, z)|| ≦ F_{1}(t)ω_{1}(l||z||α) ≦ F_{1}(t)ω_{1}(k||z||_{α}),

(12)  ||g_{2}(t, y)|| ≦ F_{2}(t)ω_{2}(Rl||y||_{C_{α}}) ≦ F_{2}(t)ω_{2}(k||y||_{C_{α}})

where k = max{l, Rl}. If we assume

∫_{0}^{∞} \frac{τ^{q−1}dσ}{ω(σ)} = ∞,

where ω = ω_{1}^{q} + ω_{2}^{q}, q = \frac{1}{β} + ϵ then also

∫_{0}^{∞} \frac{τ^{q−1}dσ}{ω(kσ)} = ∞,

and Theorem 4 yields that the solution x(t) of the boundary value problem has the property that if 0 < T < ∞ then lim_{t→T^{−}} ||x(t)||_{α} < ∞ i. e. any nonextendable solution of the boundary value problem is global.

We have proved the following theorem.

Theorem 5. Let the operator A be as above and assume that the mapping f from the right-hand side of the diffusion equation has the form f(t, z, y) = f_{1}(t, z) + f_{2}(t, y) and the following conditions are satisfied:

(1)

||f_{1}(t, z)|| ≦ F_{1}(t)ω_{1}(||z||),
||f_{2}(t, y)|| ≦ F_{2}(t)ω_{2}(||y||_{C}),

for t ≧ 0, (z, y) ∈ R^{n} × C(⟨−r, 0⟩ × Ω, R^{n}), where ω_{1}, ω_{2} : ⟨0, ∞) → (0, ∞) are continuous, positive and nondecreasing functions on R_{+}.

(2)

∫_{0}^{∞} \frac{τ^{q−1}dτ}{ω_{1}(τ)^{q} + ω_{2}(τ)^{q}} = ∞,

where q = \frac{1}{β} + ϵ, ϵ > 0, \frac{3}{4} < α = 1 − β < 1.

Then any nonextendable solution of the boundary value problem (9) is global. EJQTDE Proc. 8th Coll. QTDE, 2008 No. 13, p. 9 Acknowledgements This work was supported by the Grant No. 1/2001/05 of the Slovak Grant

Agency VEGA-SAV-MŠ.

The author would like to express sincerely thanks to the anonymous referee for his invaluable corrections, comments and suggestions.

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(Received August 31, 2007)