Ryo Ikehata

Critical exponents for semilinear dissipative wave equations in RN

Abstract We shall present new critical exponents 1+2m/N with m∈[1,2] to the Cauchy problem utt−Δu+ut=|u|p−1u with the initial data [u 0 ,u 1 ]∈(H 1 ( R N )∩L m ( R N ))×(L 2 ( R N )∩L m ( R N )) ; that is, the small data global existence property can be derived to the Cauchy problem above with power 1+2m/N

Diffusion phenomenon for linear dissipative wave equations in an exterior domain

Abstract Under the general condition of the initial data, we will derive the crucial estimates which imply the diffusion phenomenon for the dissipative linear wave equations in an exterior domain. In order to derive the diffusion phenomenon for dissipative wave equations, the time integral method which was developed by Ikehata and Matsuyama (Sci. Math. Japon. 55 (2002) 33) plays an effective role.

NEW DECAY RATES FOR A PROBLEM OF PLATE DYNAMICS WITH FRACTIONAL DAMPING

In this paper, we obtain decay rates for the total energy associated to the linear plate equation with effects of rotational inertia and a fractional damping term depending on a number θ ∈ [0, 1]. We observe that the dissipative structure of the equation with θ = 0 is of the regularity-loss type. This decay structure still remains true in the plate equation with a power of fractional damping θ > 0, but it becomes more weak when θ increase. This means that we can have an optimal decay estimate of solutions under an additional regularity assumption on the initial data. Our results generalize previous results by Luz and Charao and some of recent results due to Sugitani and Kawashima. We use a special method in the Fourier space which we developed in a previous work for the wave equation. So, our approach shows to be very effective to study decay properties for several problems in Rn.

Asymptotic profile of solutions for wave equations with frictional and viscoelastic damping terms

We consider the Cauchy problem in R for some types of damped wave equations. We derive asymptotic profiles of solutions with weighted L1,1(Rn) initial data by employing a simple method introduced in [Math. Meth. Appl. Sci. 27 (2004), 865–889].

Asymptotic profile of solutions for some wave equations with very strong structural damping

We consider the Cauchy problem in R^n for some types of damped wave equations. We derive asymptotic profiles of solutions with weighted L^{1,1}(R^n) initial data by employing a simple method introduced by the first author. The obtained results will include regularity loss type estimates, which are essentially new in this kind of equations.