Energy Decay of Solutions to the Wave Equations with Linear Dissipation Localized Near Infinity Klein-Gordon equations is studied in the Sobolev space Hs = Hs(

Energy decay for the linear and semilinear wave equations in exterior domains with some localized dissipations

“…Of course, our rsults are also valid for the case Ω = R N , the Cauchy problem, where the boundary condition of (1.2) is dropped. In order to derive the estimates of solutions for linear equations we combine some techniques in recent papers Mochizuki and Nakazawa [4] and the present author [8]. In [4], Mochizuki and Nakazawa have derived L 2 boundedness and energy decay for the case that V is star-shaped and a(x) is uniformly positive near infinity ( Hyp.A,(2)).…”

“…In order to derive the estimates of solutions for linear equations we combine some techniques in recent papers Mochizuki and Nakazawa [4] and the present author [8]. In [4], Mochizuki and Nakazawa have derived L 2 boundedness and energy decay for the case that V is star-shaped and a(x) is uniformly positive near infinity ( Hyp.A,(2)). In fact, they have treated more general dissipation which also depends on the time t. Mochizuki [3] has considered the Cauchy problem in the whole space of the quasilinear wave equation of Kirchhoff type with such a localized dissipation.…”

“…In both of papers [4] and [8] so-called multiplier methods are employed and hence, it is natural to combine these to consider the exterior problem (1.3)-(1.4) under Hyp.A . Similar techniques have been already used in E.Zuazua [15], where some semilinear wave equations in bounded domains are treated.…”

Non decay of the total energy for the wave equation with the dissipative term of spatial anisotropy

“…Mochizuki-Nakazawa [7] considers the equation in an exterior domain Ω (⊂ R N , N ≥ 3) with a smooth boundary star-shaped with respect to the origin x = 0 (where the boundary condition is the Dirichlet one). They assume that there exist positive constants R, b 0 and…”

“…Therefore a question naturally arises whether the total energy decays or not as t tends to infinity. The decay and non-decay problems have been studied by many authors, e.g., Matsumura [2], Rauch-Taylor [9], Mochizuki [3], [4], [5] and Mochizuki-Nakazawa [6], [7], Nakazawa [8] etc. These studies have clarified precisely relation between the decay property of the solutions and the decreasing condition of the coefficient b(x) when the condition is uniform with respect to spatial directions.…”