scholarly journals Global small data solutions for semilinear waves with two dissipative terms

2021 ◽  
Author(s):  
Wenhui Chen ◽  
Marcello D’Abbicco ◽  
Giovanni Girardi
Keyword(s):  
Wave Equations ◽  
Space Dimension ◽  
Full Range ◽  
Small Data ◽  
Decay Estimates ◽  
Time Decay ◽  
Nonexistence Result ◽  
Derivative Type ◽  
Long Time ◽  
Dissipative Terms

AbstractIn this work, we prove the existence of global (in time) small data solutions for wave equations with two dissipative terms and with power nonlinearity $$|u|^p$$ | u | p or nonlinearity of derivative type $$|u_t|^p$$ | u t | p , in any space dimension $$n\geqslant 1$$ n ⩾ 1 , for supercritical powers $$p>{\bar{p}}$$ p > p ¯ . The presence of two dissipative terms strongly influences the nature of the problem, allowing us to derive $$L^r-L^q$$ L r - L q long time decay estimates for the solution in the full range $$1\leqslant r\leqslant q\leqslant \infty $$ 1 ⩽ r ⩽ q ⩽ ∞ . The optimality of the critical exponents is guaranteed by a nonexistence result for subcritical powers $$p<{\bar{p}}$$ p < p ¯ .

scholarly journals Small-data shock formation in solutions to 3D quasilinear wave equations: An overview

2016 ◽  
Vol 13 (01) ◽  
pp. 1-105 ◽  
Author(s):  
Gustav Holzegel ◽  
Sergiu Klainerman ◽  
Jared Speck ◽  
Willie Wai-Yeung Wong
Keyword(s):  
Wave Equations ◽  
Initial Conditions ◽  
Large Body ◽  
Small Data ◽  
Shock Formation ◽  
Time Behavior ◽  
Long Time ◽  
Remarkable Result ◽  
Quasilinear Wave Equations ◽  
Three Space

In his 2007 monograph, Christodoulou proved a remarkable result giving a detailed description of shock formation, for small [Formula: see text]-initial conditions (with [Formula: see text] sufficiently large), in solutions to the relativistic Euler equations in three space dimensions. His work provided a significant advancement over a large body of prior work concerning the long-time behavior of solutions to higher-dimensional quasilinear wave equations, initiated by John in the mid 1970’s and continued by Klainerman, Sideris, Hörmander, Lindblad, Alinhac, and others. Our goal in this paper is to give an overview of his result, outline its main new ideas, and place it in the context of the above mentioned earlier work. We also introduce the recent work of Speck, which extends Christodoulou’s result to show that for two important classes of quasilinear wave equations in three space dimensions, small-data shock formation occurs precisely when the quadratic nonlinear terms fail to satisfy the classic null condition.

Estimates of damped fractional wave equations

2019 ◽  
Vol 22 (4) ◽  
pp. 990-1013
Author(s):  
Jianmiao Ruan ◽  
Dashan Fan ◽  
Chunjie Zhang
Keyword(s):  
Hardy Spaces ◽  
High Frequency ◽  
Wave Equations ◽  
Atomic Decomposition ◽  
Natural Extension ◽  
Decay Estimates ◽  
Lizorkin Space ◽  
Long Time ◽  
Fractional Wave Equations ◽  
Time Linear

Abstract In this paper, for the high frequency part of the solution u(x, t) to the linear fractional damped wave equation, we derive asymptotic-in-time linear estimates in Triebel-Lizorkin spaces. Thus we obtain long time decay estimates in real Hardy spaces Hp for u(x, t). The obtained results are natural extension of the known Lp estimates. Our proof is based on some basic properties of the Triebel-Lizorkin space, as well as an atomic decomposition introduced by Han, Paluszynski and Weiss.

scholarly journals Quasi-stability and continuity of attractors for nonlinear system of wave equations

2021 ◽  
Vol 8 (1) ◽  
pp. 27-45
Author(s):  
M. M. Freitas ◽  
M. J. Dos Santos ◽  
A. J. A. Ramos ◽  
M. S. Vinhote ◽  
M. L. Santos
Keyword(s):  
Wave Equations ◽  
Coupled System ◽  
Global Attractors ◽  
Time Behavior ◽  
Exponential Attractors ◽  
Small Perturbations ◽  
Recent Approach ◽  
Nonlinear Coupled System ◽  
Long Time ◽  
External Forces

Abstract In this paper, we study the long-time behavior of a nonlinear coupled system of wave equations with damping terms and subjected to small perturbations of autonomous external forces. Using the recent approach by Chueshov and Lasiecka in [21], we prove that this dynamical system is quasi-stable by establishing a quasistability estimate, as consequence, the existence of global and exponential attractors is proved. Finally, we investigate the upper and lower semicontinuity of global attractors under autonomous perturbations.

scholarly journals Correction to: On Traveling Solitary Waves and Absence of Small Data Scattering for Nonlinear Half-Wave Equations

2021 ◽  
Vol 383 (2) ◽  
pp. 1291-1294
Author(s):  
Jacopo Bellazzini ◽  
Vladimir Georgiev ◽  
Enno Lenzmann ◽  
Nicola Visciglia
Keyword(s):  
Solitary Waves ◽  
Wave Equations ◽  
Small Data ◽  
Half Wave

scholarly journals Global existence and decay estimates for nonlinear wave equations with space-time dependent dissipative term

2015 ◽  
Vol 12 (02) ◽  
pp. 249-276
Author(s):  
Tomonari Watanabe
Keyword(s):  
Global Existence ◽  
Nonlinear Wave ◽  
Wave Equations ◽  
Space Time ◽  
Time Dependent ◽  
Nonlinear Wave Equations ◽  
Energy Estimates ◽  
Space Region ◽  
Decay Estimates ◽  
Dissipative Term

We study the global existence and the derivation of decay estimates for nonlinear wave equations with a space-time dependent dissipative term posed in an exterior domain. The linear dissipative effect may vanish in a compact space region and, moreover, the nonlinear terms need not be in a divergence form. In order to establish higher-order energy estimates, we introduce an argument based on a suitable rescaling. The proposed method is useful to control certain derivatives of the dissipation coefficient.

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