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On a class of semilinear weakly hyperbolic equations

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Institution:

(1) Dipartimento di Matematica, Università di Bari, via E. Orabona 4, I-70125 Bari, Italy, Tel.: 0805442275, Fax: 0805963612,

Abstract:

Abstract In this paper, we deal with some global existence results for the large data smooth solutions of the Cauchy Problem associated with the semilinear weakly hyperbolic equations

$u_{tt}-a_{\lambda_1}(t)\Delta_x u=-a_{\lambda_2}(t)|u|^{p-1}u.$

Here u=u(x,t), $x\in \mathbb{R}^n$ and for λ≥ 0, a_λ≥ 0 is a continuous function that behaves as |t–t₀|^λ close to some t₀>0. We conjecture the existence of a critical exponent p_c(λ₁,λ₂,n) such that for p≤ p_c(λ₁,λ₂,n) a global existence theorem holds. For suitable λ₁,λ₂,n, we recall some known results and add new ones. Keywords: Critical exponents for semilinear equations, Weak hyperbolicity

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