GEORGIAN MATHEMATICAL JOURNAL: Vol. 4, No. 3, 1997, 219-222

UPPER ESTIMATE OF THE INTERVAL OF EXISTENCE OF SOLUTIONS OF A NONLINEAR TIMOSHENKO

EQUATION

DRUMI BAINOV AND EMIL MINCHEV

Abstract. Solutions to the initial-boundary value problem for a nonlinear Timoshenko equation are considered. Conditions on the initial data and nonlinear term are given so that solutions to the problem under consideration do not exist for all t > 0. An upper estimate of the t-interval of the existence of solutions is obtained. An estimate of the growth rate of the solutions is given.

1. Introduction

Let Ω ⊂ ℝ^{n} be a bounded domain with boundary ∂Ω and \overline{Ω} = Ω ∪ ∂Ω. It is assumed that the divergence theorem can be applied on Ω. Let

L^{q}(Ω) = {u: ∥u∥_{q,Ω} = ( ∫_{Ω} |u(x)|^{q}dx)^{1/q} < +∞}.

Denote by \overline{u} the complex conjugate of u.

We consider the initial-boundary value problem (IBVP) for the nonlinear Timoshenko equation

u_{tt} − φ(∥∇u∥_{2,Ω}^{2})Δu + Δ^{2}u = |u|^{p−1}u, t ≥ 0, x ∈ Ω,  (1)
u(0, x) = u_{0}(x), x ∈ \overline{Ω},  (2)
u_{t}(0, x) = u_{1}(x), x ∈ \overline{Ω},  (3)
u(t, x)|_{x∈∂Ω} = 0, t ≥ 0,  (4)
Δu(t, x)|_{x∈∂Ω} = 0, t ≥ 0,  (5)

1991 Mathematics Subject Classification. 35B30, 35B35. Key words and phrases. Nonlinear Timoshenko equation, initial-boundary problem,

interval of existence, estimate of growth rate of solutions.

219

1072-947X/97/0500-0219$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1997 Plenum Publishing Corporation

where p > 1 is a constant, φ: ℝ_{+} → ℝ_{+}, ℝ_{+} = [0, +∞), u_{0}, u_{1} : \overline{Ω} → ℂ are given functions.

Equations of type (1) appear in a variety of models describing nonlinear vibration of beams. This kind of equations is the object of long-standing interest. We should mention the papers [1]–[6], as well as the monograph [7].

In the present paper we investigate the blowing up of solutions [8] of the initial boundary value problem for the nonlinear Timoshenko equation.

2. Main Results

Theorem 1. Let the following conditions hold:

1. u is a suitably smooth classical solution of the IBVP (1)–(5);

2. φ ∈ C(ℝ_{+}, ℝ_{+}) and mΦ(s) ≥ sφ(s), ∀s ≥ 0, where Φ(s) = ∫_{0}^{s} φ(k) dk,

m ≥ 1;

3. Re ∫_{Ω} u_{0}\overline{u_{1}} dx > 0;

4. E(0) = ∥u_{1}∥_{2,Ω}^{2} + Φ(∥∇u_{0}∥_{2,Ω}^{2}) + ∥Δu_{0}∥_{2,Ω}^{2} − \frac{2}{p+1} ∥u_{0}∥_{p+1,Ω}^{p+1} ≤ 0;

5. p > 2m − 1.

Then the maximal time interval of the existence [0, T_{max}) of u can be estimated by

T_{max} ≤ T_{0} = ∫_{∥u_{0}∥_{2,Ω}^{2}}^{∞} [C_{1} − 4mE(0)ξ + \frac{4C}{p + 3} ξ^{\frac{p+3}{2}} ]^{− \frac{1}{2}} dξ < +∞,

where

C_{1} = [2 Re ∫_{Ω} u_{0}\overline{u_{1}} dx]^{2} + 4mE(0)∥u_{0}∥_{2,Ω}^{2} − \frac{4C}{p + 3} ∥u_{0}∥_{2,Ω}^{p+3},

C = 2 \frac{p + 1 − 2m}{p + 1}(\frac{1}{mes Ω})^{\frac{p−1}{2}}

and if T_{max} = T_{0}, then

lim_{t→T_{0}^{−}} ∥u(t)∥_{2,Ω}^{2} = +∞

and

lim_{t→T_{0}^{−}}(T_{0} − t)∥u(t)∥2,Ω^{\frac{p−1}{2}} ≤ \frac{2}{p − 1}\sqrt{\frac{p + 3}{C}}.

Proof. We multiply both sides of equation (1) by \overline{u}, take the real parts of both sides, and integrate to obtain

\frac{1}{2} \frac{d^{2}}{dt^{2}} (∥u∥_{2,Ω}^{2})−∥u_{t}∥_{2,Ω}^{2}+φ(∥∇u∥_{2,Ω}^{2})∥∇u∥_{2,Ω}^{2}+∥Δu∥_{2,Ω}^{2} =∥u∥_{p+1,Ω}^{p+1}. (6)

On the other hand, if we multiply both sides of equation (1) by \overline{u_{t}}, take the real parts of both sides, and integrate, we get the energy identity

∥u_{t}∥_{2,Ω}^{2} + Φ(∥∇u∥_{2,Ω}^{2}) + ∥Δu∥_{2,Ω}^{2} − \frac{2}{p + 1}∥u∥_{p+1,Ω}^{p+1} = E(0).  (7)

Thus, from (6), (7) and condition 2 of the theorem we conclude that

\frac{1}{2} \frac{d^{2}}{dt^{2}} (∥u∥_{2,Ω}^{2}) ≥ ∥u_{t}∥_{2,Ω}^{2} − ∥Δu∥_{2,Ω}^{2} + ∥u∥_{p+1,Ω}^{p+1} + m∥u_{t}∥_{2,Ω}^{2} +

+ m∥Δu∥_{2,Ω}^{2} − \frac{2m}{p + 1}∥u∥_{p+1,Ω}^{p+1} − mE(0) ≥

≥ −mE(0) + \frac{p + 1 − 2m}{p + 1}∥u∥_{p+1,Ω}^{p+1} ≥

≥ −mE(0) + \frac{p + 1 − 2m}{p + 1}(\frac{1}{mes Ω})^{\frac{p−1}{2}} ∥u∥_{2,Ω}^{p+1},

where we have used the Hölder inequality in the last step.

Denote X(t) = ∥u(t, ·)∥_{2,Ω}^{2}. Then we have the differential inequality

\frac{d^{2}X}{dt^{2}} ≥ −2mE(0) + CX^{\frac{p+1}{2}} ,  (8)

and

X(0) = ∥u_{0}∥_{2,Ω}^{2}, \frac{dX}{dt} (0) = 2 Re ∫_{Ω} u_{0}\overline{u_{1}} dx.

It is easy to prove that X′(t) > 0 wherever X(t) exists. If this conclusion is false then the set Q = {t: X′(t) ≤ 0} is nonempty. Denote t_{0} = inf Q. By integrating, we obtain

0 = X′(t_{0}) ≥ X′(0) − 2mE(0)t_{0} + C ∫_{0}^{t_{0}} X^{\frac{p+1}{2}} (τ) dτ > 0,

which is a contradiction. We multiply both sides of (8) by X′(t) and integrating twice we conclude that

T_{max} ≤ T_{0} = ∫_{∥u_{0}∥_{2,Ω}^{2}}^{∞} [C_{1} − 4mE(0)ξ + \frac{4C}{p + 3} ξ^{\frac{p+3}{2}} ]^{− \frac{1}{2}} dξ < +∞.

If T_{max} = T_{0} then lim_{t→T_{0}^{−}} ∥u(t)∥_{2,Ω}^{2} = +∞ and from (8) we have

T_{0} − t ≤ ∫_{X(t)}^{∞} [C_{1} − 4mE(0)ξ + \frac{4C}{p + 3} ξ^{\frac{p+3}{2}} ]^{− \frac{1}{2}} dξ ≤

≤ \sqrt{\frac{p + 3}{4C}} \frac{4}{p − 1} \frac{1}{[X(t)]^{\frac{p−1}{4}}} , t ∈ [T_{0} − ε, T_{0}),

where ε > 0 is sufficiently small. Thus we obtain

lim_{t→T_{0}^{−}}(T_{0} − t)∥u(t)∥2,Ω^{\frac{p−1}{2}} ≤ \frac{2}{p − 1}\sqrt{\frac{p + 3}{C}}.

Remark 1. When φ(s) = 1 + s we have m = 2 and p > 3.

Acknowledgements

This investigation was partially supported by the Bulgarian Ministry of Education, Science and Technologies under Grant MM–422.

References

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2. P. D’Ancona and S. Spagnolo, Nonlinear perturbations of the Kirchhoff equation. Comm. Pure Appl. Math. XLVII(1994), No. 7, 1005–1029.

3. R. W. Dickey, Free vibrations and dynamic buckling of the extensible beam, J. Math. Anal. Appl. 29(1970), 443–454.

4. L. A. Medeiros, On a new class of nonlinear wave equations. J. Math. Anal. Appl. 69(1979), 252–262.

5. L. A. Medeiros and M. Milla Miranda, Solutions for the equations of nonlinear vibrations in Sobolev spaces of fractionary order. Math. Apl. Comput. 6(1987), No. 3, 257–267.

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7. S. Timoshenko, D. H. Young, and W. Weaver, Jr., Vibration Problems in Engineering. John Wiley, New York, 1974.

8. H. Fujita, On the blowing-up of solutions of the Cauchy problem for u_{t} = Δu + u^{1+α}. J. Fac. Sci. Univ. Tokyo, Sect. IA 13(1966), 109–124.

(Received 19.02.1996)

Authors’ address: Medical University of Sofia P.O. Box 45, Sofia 1504, Bulgaria