Movement of time-delayed hot spots in Euclidean space

Abstract

We investigate the shape of the solution of the Cauchy problem for the damped wave equation. In particular, we study the existence, location and number of spatial maximizers of the solution.Studying the shape of the solution of the damped wave equation, we prepare a decomposed form of the solution into the heat part and the wave part. Moreover, as its another application, we give L p -L q estimates of the solution.

Estimates of Lifespan and Blow-up Rates for the Wave Equation with a Time-dependent Damping and a Power-type Nonlinearity

“…Then, he proved the small data global existence when p > p F and the sharp upper bound of the lifespan (1.3) when p ≤ p F . For n = 1, 2 and n ≥ 4, the same type decomposition was obtained by Marcati and Nishihara [20], Hosono and Ogawa [13] and Narazaki [23] (see also Sakata and the third author [27] for the exact decomposition for n ≥ 4).…”

Diffusion phenomena for the wave equation with space-dependent damping in an exterior domain

“…Indeed, for the damped wave equation with constant damping in the whole space u tt − ∆u + u t = 0, x ∈ R N , t > 0, (1.6) many mathematicians studied the asymptotic behavior of solutions and verified the diffusion phenomena. We refer the reader to [22,4,31,32,53,14,33,21,5,30,44]. For an exterior domain Ω ⊂ R N , namely, in the case a(x) ≡ 1 in our problem (1.1), the diffusion phenomena was proved by [8,11,2,40].…”

Movement of time-delayed hot spots in Euclidean space for a degenerate initial state

“…It is one of the most fundamental properties of the damped wave equation that, as t goes to infinity, the unique classical solution of (1.1) approaches to that of the corresponding heat equation, P n (t)(f +g)(x) = 1 (4πt) n/2 R n exp − |x − y| 2 4t (f +g)(y)dy, x ∈ R n , t > 0, (1.2) if f + g does not vanish. Such a property is called the diffusion phenomenon and has been studied by several researchers [2,3,8,7,9,11,12,13]. We also refer to [4] for the diffusion phenomenon in an exterior domain.…”

“…In this paper, we study the equation (1.1) in the latter sense. We refer to [6] for the deriving process as the heat equation with finite propagation speed (see also [12,Introduction]).…”