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Blow up of solutions of a generalized Boussinesq equation

Authors:	De Godefroy A

Institution:	Laboratoire d'Analyse Numerique, Centre d'Orsay, Universite Paris XI, France ^Y Present address: Centre Universitaire de Cocody, a la FAST, Departement de Mathematiques, BP 582, 22 Abidjan, Ivory Coast

Abstract:	Consider the Cauchy problem u_tt = (f(u))_xx + u_xxtt x $BORDER=$ R, t $≥$ 0, u(x,0) = u₀(x), u_t(x,0) = u₁(x),7rcub; where f : R $->$ R C, f(0) = ). After treatment of the local existenceproblem, we show the blow up of the solution of the equation(1) under the folowing assumptions. Let ${alpha}$ > 0 be real, such that 2(l + 2 ${alpha}$ )F(u) $≥$ uf(u), $1/2;$ (v₀, Pv₀)_l² + ${int}$ _- F(u₀)dx < 0 where P = 1 - d²/dx², F'(s), and v₀ is given by u₁(x,0) = (v₀(x))_x. Then we focus on various perturbations of the question. We alsostudy the vectorial case in the same way, and finally we giveexamples.

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