Hyperbolic-like estimates for higher order equations

Abstract

a b s t r a c tThe main goal of this paper is to derive long time estimates of the energy for the higher order hyperbolic equations with time-dependent coefficients. In particular, we estimate the energy in the hyperbolic zone of the extended phase space by means of a function f (t) which depends on the principal part and on the coefficients of the terms of order m−1. Then we look for sufficient conditions that guarantee the same energy estimate from above in all the extended phase space. We call this class of estimates hyperbolic-like since the energy behavior is deeply depending on the hyperbolic structure of the equation. In some cases, these estimates produce a dissipative effect on the energy.

A class of dissipative wave equations with time-dependent speed and damping

“…Since (t) = 1 + log Λ(t), it is easy to check that condition (13) holds. We notice that (16) does not hold in this case.…”

“…If κ ∈ (0, 1] then (16) holds, hence Remark 6 is applicable and (13) is satisfied since from κ ≤ 1 it follows that…”

The threshold of effective damping for semilinear wave equations

“…has been deeply investigated. In particular, if we assume small, compactly supported data, then by using some linear decay estimates [17] one can prove that there exists a global solution to (1) if p > 1 + 2/n, and p ≤ 1 + 2/(n − 2) if n ≥ 3 (see [22]). This exponent is critical, that is, for suitable nontrivial, arbitrarily small data and f (u) = |u| p with 1 < p ≤ 1 + 2/n, there exists no global solution to (1) (see [22,31]).…”

“…In particular, if we assume small, compactly supported data, then by using some linear decay estimates [17] one can prove that there exists a global solution to (1) if p > 1 + 2/n, and p ≤ 1 + 2/(n − 2) if n ≥ 3 (see [22]). This exponent is critical, that is, for suitable nontrivial, arbitrarily small data and f (u) = |u| p with 1 < p ≤ 1 + 2/n, there exists no global solution to (1) (see [22,31]). If one removes the compactness assumption on the data, still one may obtain global existence for p > 1 + 2/n if the data are small in the norm of the energy space (H 1 × L 2 ) and in the L 1 norm in space dimension n = 1, 2 (see [9]).…”

“…For n = 3, the range of admissible exponents p for the global existence in Theorem 2 is nonempty if, and only if, γ < 4. We have the range (1 + (2 + γ)/3, 3] if γ ∈ [1,4), and the range [2,3]…”