Long Time Decay Estimates in Real Hardy Spaces for the Double Dispersion Equation

2020 ◽  
pp. 181-190
Author(s):  
Marcello D’Abbicco ◽  
Alessandra De Luca
Keyword(s):  
Dispersion Equation ◽  
Hardy Spaces ◽  
Decay Estimates ◽  
Time Decay ◽  
Long Time ◽  
Double Dispersion

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 ¯ .

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 Long-Time Decay to the Global Solution of the 2D Dissipative Quasigeostrophic Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Jamel Benameur ◽  
Mongi Blel
Keyword(s):  
Global Solution ◽  
Time Decay ◽  
Long Time

We study the behavior at infinity in time of any global solutionθ∈C(R+,Ḣ2-2α(R2))of the surface quasigeostrophic equation with subcritical exponent2/3≤α≤1. We prove thatlim⁡t→∞∥θ(t)∥Ḣ2-2α=0. Moreover, we prove also the nonhomogeneous version of the previous result, and we prove that ifθ∈C(R+,Ḣ2-2α(R2))is a global solution, thenlim⁡t→∞∥θ(t)∥H2-2α=0.

scholarly journals Global existence of solution for multidimensional generalized double dispersion equation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Xiaoqiang Dai
Keyword(s):  
Cauchy Problem ◽  
Global Existence ◽  
Dispersion Equation ◽  
Existence Of Solutions ◽  
Existence Of Solution ◽  
New Methods ◽  
Global Existence Of Solutions ◽  
The Cauchy Problem ◽  
Double Dispersion

Abstract In this paper, we study the Cauchy problem of multidimensional generalized double dispersion equation. To prove the global existence of solutions, we introduce some new methods and ideas, and fill some gaps in the established results.

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