Nonexistense of global solutions of a quasilinear bi-hyperbolic equation with dynamical boundary conditions

V. Bayrak, Istanbul Technical University, Istanbul, TURKEY
M. Can, Istanbul Technical University, Istanbul, TURKEY

E. J. Qualitative Theory of Diff. Equ., No. 3. (1999), pp. 1-10.

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 system of equations. 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.

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