$L^p-L^q$ estimate for wave equation with bounded time dependent coefficient

The influence of oscillations on global existence for a class of semi-linear wave equations

“…under the following assumptions: For the shape function: The smooth function = (t) satisfies (9) and (10).…”

Diagonalisation schemes and applications

“…The construction forms the core of the more involved multi-step scheme introduced later in Section 2.2 and is the key idea in the diagonalisation-based approaches of Yagdjian, Reissig and co-authors, see e.g., [23], [13], [9], [8]. For more detailed discussions on applications we refer to Section 3.…”

“…In this case one has to be careful by choosing t large in dependence of the parameter and this leads to the introduction of so-called zones. We will not go into details here, but refer the reader to the fundamental treatise of Yagdjian on weakly hyperbolic problems, [23,Chapter 3], and applications deducing dispersive estimates for wave equations with variable propagation speed of Reissig and co-authors, [13], [9]. More involved considerations including several zones and different diagonalisation hierarchies turn up for the treatment of lower-order terms, e.g.…”

On the asymptotic behavior of the energy for the wave equations with time depending coefficients

“…Thus the necessity of the condition a (t) ∈ L 1 ((0, ∞)) for the estimate (3) arises as a natural question. A partial answer to this question has been proposed by Reissig and Smith [11], and their properties are represented as follows:…”

“…Then, it may be a natural expectation that C m properties of a(t) can relax the restriction to a (t) further for (3) as m becomes larger, but the following result from [11] contradicts such a expectation:…”