A blow-up result for a nonlinear wave equation with damping: The critical case

By Qi S. Zhang. 2001

347 citations recorded by the source corpus.

Subfields: Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b4f673ce3cb355ed75601143ad3b00c8b7ed9c19

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In Control and Systems Engineering

In Mathematical Physics

Citations in the corpus

Blow Up of Solutions to Semilinear Wave Equations with Critical Exponent in High Dimensions* (2007, result): “Then we can get a sharp lower bound of certain averaged quantities which depends on T. The new idea is to get an upper bound of the averaged quantities which is independent of T. In this way, we can get an upper bound for T. This idea was inspired by the recent work of Q. S. Zhang [ 11 ] on the damp”
The Threshold between Effective and Noneffective Damping for Semilinear Waves (2012, background): “This exponent is critical, that is, for suitable nontrivial, arbitrarily small data and f(u) = |u| with 1 < p ≤ 1+ 2/n, there exists no global solution to (1) (see [22, 31]).”
A competition between Fujita and Strauss type exponents for blow-up of semi-linear wave equations with scale-invariant damping and mass (2019, methodology): “e quadratic equation (n−1)p2 −(n+1)p−2 = 0. In particular, we refer to the classical works [16, 17, 8, 27, 26, 15, 39, 41] and references therein for blow-up results when 1 <p≤ p 0(n). In [30] and [40] the massless case with constant coeﬃcients in the linear part is considered. While in [30] the ”
L^P-L^Q Estimates for Damped Wave Equations and their Applications to Semi-Linear Problem (2004, background): “Zhang [31] have studied the critical case s 1⁄4 scðnÞ and he have shown the non-existence of time global solution to (1.”
Small data blow-up of $L^2$-solution for the nonlinear Schrödinger equation without gauge invariance (2011, methodology): “∆v+vt = |v|p, (t,x) ∈ R ×Rn, v(0,x) = v0(x),vt(0,x) = v1(x), x∈ Rn, where 1 < p ≤ 1 + 2/n,v = v(t,x) is a real-valued unknown function, v0(x) and v1(x) are compactly supported given functions (see [19]). However, since this method needs a positivity of the nonlinear term |v|p, it can not be appli”
New decay estimates for linear damped wave equations and its application to nonlinear problem (2004, neutral): “D whole space it is known (see References [21,24]) that the critical exponent of (6) is equal to p∗”
Blow-up for semilinear wave equations with the scale invariant damping and super-Fujita exponent (2017, methodology): “ 1 <p<1+ 2 n+µ− 1 and 0 <µ≤ 1, where Cis a positive constant independent of ε. We note that the both proofs in [12] and [13] are based on so-called “test function method” introduced by Zhang [18]. On the other hand, D’Abbicco [1] has obtained the global existence result if p>pF(n), but µ”
Semi‐linear wave models with power non‐linearity and scale‐invariant time‐dependent mass and dissipation (2017, methodology): “4 In the proof of this theorem we will use the test function method, see, for example, [5], [23] and [28].”
Damped wave equation with a critical nonlinearity (2005, background): “This blow-up result was extended to the critical and subcritical cases 1 < ρ ≤ 1 + 2 n in [19].”
Global asymptotics of solutions to the Cauchy problem for the damped wave equation with absorption (2006, background): “There have been many studies of the non-absorbing semilinear term problem [7,8,13–17,22, 24,31,32,34], etc.”
$ L^p $-$ L^q $ estimates for the damped wave equation and the critical exponent for the nonlinear problem with slowly decaying data (2017, background): “We recall that when r = 1, p = 1 + 2 n , and N (u) = |u| , the local solution blows up in a finite time even if the size of the initial data ε is arbitrary small, provided that the shape of the initial data (u0, u1) ∈ (H(1)(R)∩L(1)(R))×(L(2)(R)∩L(1)(R)) has a positive integral average (see [37]).”