M athematical I nequalities & A pplications Volume 1, Number 3 (1998), 367–374 NONEXISTENCE OF GLOBAL SOLUTIONS OF A QUASILINEAR HYPERBOLIC EQUATION VURAL BAYRAK,MEHMET CAN AND F AHREDDIN AL¯ IYEV 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 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 concavity 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 concludes that in finite time t 2 , this functional and hence the norm of the solution blows up. Mathematics subject classification (1991): 35L70, 35B05, 65M06. Key words and phrases: Quasilinear hyperbolic equations, nonexistence of global solutions, blow up of solutions. REFERENCES [1] R. T. GLASSEY, Blow-up theorems for nonlinear wave equations, Math. Z. 132 (1973), 183–203. [2] O. A. LADYZHENSKAYA AND V. K. KALANTAROV, Blow up theorems for quasilinear parabolic and hypherbolic equations, Zap. Nauckn. SLOMI Steklov 69 (1977), 77–102. [3] H. A. LEVINE, Instability and nonexistence of global solutions to nonlinear wave equations of the form Pu tt = -Au + F (u) , Trans. Am. Math. Soc. 192 (1974), 1–22. [4] J. L. LIONS, Equations Diff´ erentielles Op´ erationelles et Probl´ eme aux Limites, Springer, 1961. c  , Zagreb Paper MIA-01-35 Mathematical Inequalities & Applications www.ele-math.com mia@ele-math.com