Fumihiko Hirosawa

Generalised energy conservation law for wave equations with variable propagation speed

Abstract We investigate the long time behaviour of the L 2 -energy of solutions to wave equations with variable speed of propagation. The novelty of the approach is the combination of estimates for higher order derivatives of the coefficient with a stabilisation property.

Cm-theory of damped wave equations with stabilisation

Abstract The aim of this note is to extend the energy decay estimates from [J. Wirth, Wave equations with time-dependent-dissipation. I: Non-effective dissipation, J. Differential Equations 222 (2006) 487–514] to a broader class of time-dependent dissipation including very fast oscillations. This is achieved using stabilisation conditions on the coefficient in the spirit of [F. Hirosawa, On the asymptotic behavior of the energy for wave equations with time-depending coefficients, Math. Ann. 339 (4) (2007) 819–839].

From Wave to Klein—Gordon Type Decay Rates

The goal of the paper is to derive L p —L q decay estimates for Klein—Gordon equations with time-dependent coefficients. We explain the influence of the relation between the mass term and the wave propagation speed on L p —L q decay estimates. Contrary to the classical Klein—Gordon case we cannot expect in each case a Klein—Gordon type decay rate. One has wave type decay rates, too. Moreover, under certain assumptions no L p —L q decay estimates can be proved. In these cases the solution has a Floquet behavior. More precisely, one can show that the energy cannot be estimated from above by time-dependent functions with a suitable growth order if t tends to infinity.

Well-Posedness in Sobolev Spaces for Second-Order Strictly Hyperbolic Equations with Nondifferentiable Oscillating Coefficients

The goal of this paper is to study well-posedness to strictly hyperbolic Cauchyproblems with non-Lipschitz coefficients with low regularity with respect to timeand smooth dependence with respect to space variables. The non-Lipschitz conditionis described by the behaviour of the time-derivative of coefficients. This leads to a classification of oscillations, which has a strong influence on the loss of derivatives. To study well-posednesswe propose a refined regularizing technique. Two steps of diagonalizationprocedure basing on suitable zones of the phase spaceand corresponding nonstandard symbol classes allow to applya transformation corresponding to the effect of loss of derivatives.Finally, the application of sharp Gårdings inequality allows to derive a suitable energy estimate. From this estimatewe conclude a result about C∞well-posedness of the Cauchy problem.

Generalized energy conservation for klein–gordon type equations

Abstract The aim of this paper is to derive energy estimates for solutions of the Cauchy problem for the Klein–Gordon type equation ut t 4uCm(t)2u D 0. The coefficient m is given by m(t)2 D (t)2 C p(t) with a decreasing, smooth shape function and an oscillating, smooth and bounded perturbation function p. We study under which assumptions for and p one can expect results about a generalization of energy conservation. The main theorems of this note deal with m belonging to C M , M 2, and m belonging to the Gevrey class (s), s 1.

A class of non-analytic functions for the global solvability of Kirchhoff equation

We consider the global solvability to the Cauchy problem of Kirchhoff equation with generalized classes of Manfrins class. Manfrins class is a subclass of Sobolev space, but we shall extend this class as a subclass of the ultradifferentiable class, and we succeed to prove the global solvability of Kirchhoff equation with large data in wider classes from the previous works.

On the Global Solvability for Semilinear Wave Equations with Smooth Time Dependent Propagation Speeds

In this paper we consider the global existence of a solution with small data to the Cauchy problem for the semilinear wave equations with time dependent coefficient:

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

u_{tt} - a(t)^2 \varDelta u=u_t^2-a(t)^2| \nabla u|^2.