NONEXISTENCE OF GLOBAL SOLUTIONS OF A QUASILINEAR BI-HYPERBOLIC EQUATION WITH

DYNAMICAL BOUNDARY CONDITIONS

Vural BAYRAK and Mehmet CAN

Istanbul Technical University Mathematics Department Maslak, 80626 Istanbul TURKEY

AMS classification: 35L70, 35L60 Keywords: Blow up of solutions, bi-hyperbolic equations, nonexistence of global solutions. e-mail: mcan@itu.edu.tr

Abstract

In this work, the nonexistence of the global solutions to a class of initial boundary value problems with dissipative terms in the boundary conditions is considered for a quasilinear bi-hyperbolic equation. The nonexistence proof is achieved by the use of a lemma due to O. Ladyzhenskaya and V.K. Kalantarov and by the usage of the so called generalized convexity method. In this method one writes down a functional which reflects the properties of dissipative boundary conditions and represents the norm of the solution in some sense, then proves that this functional satisfies the hypotheses of Ladyzhenskaya-Kalantarov lemma. Hence from the conclusion of the lemma one deduces that in a finite time t_{2}, this functional and hence the norm of the solution blows up.

1 INTRODUCTION

Initial-boundary value problems written for hyperbolic quasilinear partial differential equations emerged in several applications to physics, mechanics and engineering sciences. Natures of the solutions to these equations have been investigated by several means.

The nonexistence of global solutions of quasilinear hyperbolic equations with no dissipative terms in the boundary conditions are investigated for example by J.L. Lions [1], R.T. Glassey [2], H.A. Levine [4 ,5, 6], and O.A. Ladyzhenskaya and V. K. Kalantarov [8] and many others. Levine has a survey article [7] with many relevant references (See also Straughan [9]).

In [4] Levine studied the initial value problem for the following “abstract” wave equation with dissipation

P u_{tt} + A_{1}u_{t} + Au = F (u)

in a Hilbert space where P, A_{1} and A are positive linear operators defined on some dense subspace of the Hilbert space and F is a gradient operator with potential F. It is assumed that (u, F (u)) ≥ F(u) for all u in the domain of F. The global nonexistence result he proved is the following: If the energy is initially negative then the solution can not be global. This is the same result that he proved in the case that A_{1} = 0 (See [6]).

However, the functional used for the investigation of initial-boundary value problems with no dissipative terms in the boundary conditions can not be directly applied to the problems with the dissipative terms in the boundary conditions. Whenever damping is present, one must allow for the possibility that the data restrictions could be more severe than without damping [10]-[14].

The tool used in this work is a lemma due to O.A. Ladyzhenskaya and V.K. Kalantarov [8]. Part (b) of the Lemma was introduced also by H. A. Levine in [3,6]. From now on we will call it the LK Lemma.

The most crucial point in the application of this tool is to find a functional that represents the dissipation on the boundary and satisfies the conditions of the LK Lemma. This method is known as the “generalized convexity” method.

Let us begin by stating LK Lemma [8] without proof.

LEMMA

If a function

ψ(t) ∈ C^{2}, ψ(t) ≥ 0

satisfies the inequality

ψ′′(t)ψ(t) − (1 + γ)ψ′(t)^{2} ≥ −2C_{1}ψ(t)ψ′(t) − C_{2}ψ(t)^{2}

for some real numbers γ > 0, C_{1}, C_{2} ≥ 0, then the following hold

a) If

(1)  ψ(0) > 0, ψ′(0) > −γ_{2}γ^{−1}ψ(0), C_{1} + C_{2} > 0

where

γ_{1} = −C_{1} + \sqrt{C_{1}^{2} + γC_{2}}, γ_{2} = −C_{1} − \sqrt{C_{1}^{2} + γC_{2}},

then there exists a positive real number,

t_{1} < t_{2} = \frac{1}{2\sqrt{C_{1}^{2} + C_{2}^{2}}} ln \frac{γ_{1}ψ(0) + γψ′(0)}{γ_{2}ψ(0) + γψ′(0)}

such that ψ(t) ⟶ +∞ as t ⟶ t_{1}.

b) If ψ(0) > 0, ψ′(0) > 0 and C_{1} = C_{2} = 0 then there exists a positive real number t_{1} ≤ t_{2} = ψ(0)/(γψ′(0)), such that ψ(t) ⟶ +∞ as t ⟶ t_{1}.

2 THE IBV PROBLEM

Let us now consider the initial-boundary value problem:

(2)  w_{tt} + Δ^{2}w = f(−Δw) + Δw, (t, x) ∈ (0, T) × Ω̄,

−Δw = 0, \frac{∂w_{t}}{∂ν} − Δ^{2}w = 0, (t, x) ∈ (0, T) × ∂Ω,

w(x, 0) = w_{0}(x), w_{t}(x, 0) = w_{1}(x) x ∈ Ω.

where Ω is a bounded domain in R^{n} with a sufficiently smooth boundary ∂Ω := Γ, T > 0 is an arbitrary real number, and ν is the outward normal of the boundary Γ.

Let us define a functional ψ(t) by

(3)  ψ(t) = ∫_{Ω} |∇w|^{2}dx + ∫_{0}^{t} ∫_{Γ} (\frac{∂w}{∂ν} )^{2} dxdt + ∫_{Γ} (\frac{∂w_{0}}{∂ν} )^{2} dx,

and let (4) E(0) = ∫_{Ω} |∇w_{1}|^{2}dx + ∫_{Ω} |∇Δw_{0}|^{2}dx + ∫_{Ω}(Δw_{0})^{2}dx − 2 ∫_{Ω} F(Δw_{0})dx ≤ 0.

be the initial energy of the system. Then by the use of the LK Lemma, the following theorem will be proved:

EJQTDE, 1999 No. 3, p. 5

THEOREM. Let the function f(u) with its primitive F(u) = ∫^{u} f(s)ds have the following properties:

(5)  f(0) = 0, sf(s) ≥ 2(2γ + 1)F(s), ∀s ∈ R^{1}

for some real number γ > 0, and let w_{0}(x) and w_{1}(x) be two functions such that

1) For ψ in the above (3), and its derivative ψ′, the inequality (1) in LK Lemma holds.

2) The initial energy E(0) in (4) is nonpositive.

If t_{2} > 0 is the number given in the LK Lemma, then there exists a positive real number t_{1} < t_{2} such that ψ(t) ⟶ +∞ as t ⟶ t_{1}.

PROOF. Differentiating (3) with respect to t one has

(6) ψ′(t) = 2 ∫_{Ω} ∇w · ∇w_{t}dx + 2 ∫_{0}^{t} ∫_{Γ} \frac{∂w}{∂ν} \frac{∂w_{t}}{∂ν} dxdt + ∫_{Γ} (\frac{∂w_{0}}{∂ν} )^{2} dx.

A further differentiation with respect to t gives

ψ′′(t) = 2 ∫_{Ω} |∇w_{t}|^{2}dx + 2 ∫_{Ω} ∇w · ∇w_{tt}dx + 2 ∫_{Γ} \frac{∂w}{∂ν} \frac{∂w_{t}}{∂ν} dx.

Using the Green-Gauss theorem, the partial differential equation in (2) on the boundary, we convert the second and the third integrals in the above to get

ψ′′(t) = 2 ∫_{Ω} |∇w_{t}|^{2}dx − 2 ∫_{Ω}(Δw)w_{tt} dx.

Substituting w_{tt} as in (2) and using the inequality in (5) one obtains

ψ′′(t) ≥ 2 ∫_{Ω} |∇w_{t}|^{2}dx − 2 ∫_{Ω} |∇Δw|^{2}dx−

(7)  2 ∫_{Ω}(Δw)^{2}dx + 4(2γ + 1) ∫_{Ω} F(−Δw)dx.

To make a better estimate for ψ′′(t), let us multiply both sides of the equation in (2) by −2Δw_{t}, and integrate over Ω to get

\frac{∂E(t)}{∂t} = −2 ∫_{Γ} (\frac{∂w_{t}}{∂ν} )^{2} dx

where

E(t) = ∫_{Ω} |∇w_{t}|^{2}dx + ∫_{Ω} |∇Δw|^{2} dx + ∫_{Ω}(Δw)^{2}dx − 2 ∫_{Ω} F(−Δw)dx

can be regarded as the total energy of the system.

Hence

E(t) = E(0) − 2 ∫_{0}^{t} ∫_{Γ} (\frac{∂w_{t}}{∂ν} )^{2} dxdt

¿From the second hyphothesis of the theorem, the initial energy E(0) is nonpositive. Therefore E(t) < 0, ∀t > 0.

Adding

2(2γ + 1) (E(t) − E(0) + 2 ∫_{0}^{t} ∫_{Γ} (\frac{∂w_{t}}{∂ν} )^{2} dxdt) = 0

to the right hand side of (7) and omitting some of the positive terms one gets the estimate

(8)  ψ′′(t) ≥ 4(γ + 1) (∫_{Ω} |∇w_{t}|^{2} dx + a ∫_{0}^{t} ∫_{Γ} (\frac{∂w_{t}}{∂ν} )^{2} dxdt) .

If we summarize,

ψ(t) = A1 + B1 + C, ψ′(t) = 2 A2 + 2 B2 + C, ψ′′(t) ≥ 4(γ + 1)(A3 + B3), where A1, B1, C, A2, B2, A3, B3 represent the corresponding integrals in (3), (6) and (8).

In LK lemma

ψ′′(t)ψ(t) − (1 + γ)[ψ′(t)]^{2} ≥
4(1 + γ)[(A3 + B3)(A1 + B1 + C) − (A2 + B2 + C/2)^{2}] ≥
4(1 + γ)[H − ψ(t)ψ′(t)/2],

where

H = (A3 + B3)(A1 + B1) − (A2 + B2)^{2} ≥ 0

by Cauchy-Schwarz inequality. Therefore the hypotheses of LK Lemma are satisfied with C_{1} = 1 + γ, and C_{2} = 0. Hence from the conclusion of the Lemma, the Theorem is proved.

ACKNOWLEDGMENT

We are grateful for many illuminating discussions with M. Kirane.

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