Solution representations for a wave equation with weak dissipation

By Jens Wirth. 2002

159 citations recorded by the source corpus.

We consider the Cauchy problem for the weakly dissipative wave equation □v+μ/1+t vt=0, x∈ℝn, t≥0 parameterized by μ>0, and prove a representation theorem for its solutions using the theory of special functions.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7eaa0ba49221d1e5f53ee99718029b358aef10f2

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

A note on the blowup of scale invariant damping wave equation with sub-Strauss exponent (2017, background): “For this typical damping case, the asymptotic behavior of linear equation heavily relies on the size of μ see [11].”
The Threshold between Effective and Noneffective Damping for Semilinear Waves (2012, methodology): “To prove them we follow the approach used in [25] to derive L(2) − L(2) estimates for the linear damped wave equation utt −△u+ μ 1 + t ut = 0 , t ≥ 0 , modifying it to derive (L∩L(2))−L(2) estimates, and taking into account the presence of the parameter s and of the speed λ(t).”
Blow-up for semilinear wave equations with the scale invariant damping and super-Fujita exponent (2017, methodology): “(x), x∈ Rn, (1.1) where µ>0,f,g∈ C∞ 0 (R n) and n∈ N. We assume that ε>0 is a “small” parameter. First, we shall outline a background of (1.1) brieﬂy according to the classiﬁcations by Wirth in [20, 21, 22] for the corresponding linear problem. Let u0 be a solution of the initial value problem”
Diffusion phenomenon in Hilbert spaces and applications (2011, background): “• Reference [39], [41], [42] had the page ranges corrected.”
Semi‐linear wave models with power non‐linearity and scale‐invariant time‐dependent mass and dissipation (2017, methodology): “using the recurrence formulas for the derivatives of the Hankel functions νH± ν (z) + z(H± ν )′(z) = zH± ν−1(z), we have for any j, k = 1, 0 ∂ t j (t, s, ξ) = (−1) j iπ 4 (1 + t) (1 + s)ρ−1 |ξ | 1+k− j ∣∣∣∣H − ρ−1+ j ((1 + s)|ξ |) H− ρ−k((1 + t)|ξ |) H+ ρ−1+ j ((1 + s)|ξ |) H+ ρ−k((1 + t)|ξ |) ∣∣∣∣ ”
A shift in the Strauss exponent for semilinear wave equations with a not effective damping (2015, background): “ such that there exist no global (in time) solutions if p∈ (1,pcrit]. It has been recently shown that the critical exponent for models with eﬀective dissipation, this means, µis suﬃciently large (see [27]) vtt −△v+ µ 1+t vt = |v|p, t≥ 0, x∈ Rn, v(0,x) = v0(x), x∈ Rn, vt(0,x) = v1(x), x∈ Rn, (3)”
Life-span of semilinear wave equations with scale-invariant damping: Critical Strauss exponent case (2017, background): “For this typical damping, the asymptotic behavior of linear solution heavily relies on the size of μ see [16].”
The threshold of effective damping for semilinear wave equations (2015, methodology): “Combining (24) and (26) with D 0 and D 1, and s D 0, we obtain the energy estimates derived in [15].”
A classification of structural dissipations for evolution operators (2016, background): “t/ 2 Œ1,1/ is more delicate (see [5, 6]), whereas for limt!1 tb.”
Weighted L2-estimates for dissipative wave equations with variable coefficients (2009, unknown): “In contrast to the enormous number of works concerning L2-estimates, the number of general techniques is quite small: (1) resolvent estimates for parameter dependent elliptic operators [AL], [CZ], [FZ], [HM], [RT1], [RW], [SJ], [ W1 ], (2) observability inequalities combined with the energy dissipat”