JournalsrlmVol. 22, No. 2pp. 135–149

Propagation of analyticity for a class of nonlinear hyperbolic equations

  • Sergio Spagnolo

    Università di Pisa, Italy
Propagation of analyticity for a class of nonlinear hyperbolic equations cover
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Abstract

We consider the semilinear hyperbolic equations of the form

tmu+,a1(t)tm1xu+,+,am(t)xmu=f(u)\partial_{t}^mu \,+, a_1(t)\,\partial_{t}^{m-1}\partial_{x} u \,+, \cdots \,+, a_m(t)\,\partial_{x}^mu \,=\, f(u)

with f(u)f(u) entire analytic, where the characteristic roots satisfy

λi2(t),+,λj2(t)M(λi(t)λj(t))2,  ij.\lambda_i^2(t),+,\lambda_j^2(t) \le M( \lambda_i(t)-\lambda_j(t))^2 \,, \quad \hbox{} \ \ i\neq j.

We prove that, if the ah(t)\,a_h(t)'s are analytic functions, al the solutions bounded in C\mathcal {C}^\infty enjoy the propagation of analyticity; while, if the ah(t)a_h(t)'s are C\mathcal {C}^\infty functions, such a property holds for those solutions which are bounded in some Gevrey class.

Cite this article

Sergio Spagnolo, Propagation of analyticity for a class of nonlinear hyperbolic equations. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 22 (2011), no. 2, pp. 135–149

DOI 10.4171/RLM/591