Selfsimilar profiles in large time asymptotics of solutions to damped wave equations

By Grzegorz Karol Karch. 2000

171 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/cb7bfd2d17d1218c223805df550f76534efd6b81

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Citations in the corpus

A classification of structural dissipations for evolution operators (2016, background): “0, 1=2/, the diffusion phenomenon proved in [1] has been extended to the general Lp Lq setting, 1 p q 1, in [14].”
On the asymptotic behavior of solution for the generalized IBq equation with hydrodynamical damped term (2012, methodology): “We refer to [10, 19,21,26] for damped wave equations and [9,14,22] for wave equations of memory-type dissipation.”
Damped wave equation with a critical nonlinearity (2005, background): “Applying energy type estimates obtained in [11] and [8] it was proved in [7] that solutions of the nonlinear damped wave equation (1.”
$ L^p $-$ L^q $ estimates for the damped wave equation and the critical exponent for the nonlinear problem with slowly decaying data (2017, background): “Based on the linear estimates (3), many mathematicians studied the global existence and the blow-up of solutions (see [9, 10, 11, 12, 13, 15, 16, 18, 19, 20, 21, 22, 23, 24, 27, 30, 34, 36, 37] and the references therein).”
Decay properties of solutions to the Cauchy problem for the damped wave equation with absorption (2006, background): “Based on [21], Karch [20] showed that the Gauss kernel is an asymptotic profile when p > 1 + 4/N with 1 N 3.”
Diffusion phenomenon for second order linear evolution equations (2003, background): “Furthermore, in [6] Karch has discovered the asymptotic self-similarity as t → ∞ of solutions to the equation (1.”
Global existence and asymptotics of solutions of the Cahn–Hilliard equation (2007, background): “Differential Equations 238 (2007) 426–469 The next lemma is concerned with the asymptotics of T (t)u0 with u0(x) ∈ L1(RN,R) whose proof can be found in [18,19] (cf.”
A new phenomenon in the critical exponent for structurally damped semi-linear evolution equations (2017, background): “However, when 2 δ ∈ (0 , σ ), the asymptotic profile for ∥ u ( t , · ) ∥ L q and ∥ u t ( t , · ) ∥ L q , where u is the solution to the linear problem (6), changes whether condition holds or not (see [7, 28] and, later, Proposition 4.1).”
Energy decay estimates for wave equations with a fractional damping (2012, background): “These properties (at least) in the case of θ ∈ (0, 1/2) and θ = 1 have already been observed due to Karch [13] and Mugnolo [18], respectively.”
Damped wave equation with super critical nonlinearities (2004, methodology): “Applying energy type estimates obtained in papers [6] and [4] it was proved in [3] that solutions of the nonlinear damped wave equation (2) in the super critical cases 1+ 4 n < ρ ≤ n n−2 , if n = 3 and 1 + 4 n < ρ < ∞, if n = 1, 2, with arbitrary initial data u0 ∈ H(1) ∩ L(1), u1 ∈ L(2) ∩ L(1) (i.”
Diffusion phenomenon in Hilbert spaces and applications (2011, methodology): “These estimates combined with the Nash inequality and Markov property for the parabolic semigroup yield explicit and sharp decay rates for the hyperbolic problem.”
Global asymptotics of solutions to the Cauchy problem for the damped wave equation with absorption (2006, background): “1) under the assumption in [19].”