Long time decay estimates in real Hardy spaces for evolution equations with structural dissipation

2015 ◽  
Vol 7 (2) ◽  
pp. 261-293 ◽  
Author(s):  
M. D’Abbicco ◽  
M. R. Ebert ◽  
T. Picon
Author(s):  
Wenhui Chen ◽  
Marcello D’Abbicco ◽  
Giovanni Girardi

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 ¯ .


2019 ◽  
Vol 22 (4) ◽  
pp. 990-1013
Author(s):  
Jianmiao Ruan ◽  
Dashan Fan ◽  
Chunjie Zhang

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.


2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Yongqin Xie ◽  
Zhufang He ◽  
Chen Xi ◽  
Zheng Jun

We prove the asymptotic regularity of global solutions for a class of semilinear evolution equations in H01(Ω)×H01(Ω). Moreover, we study the long-time behavior of the solutions. It is proved that, under the natural assumptions, these equations possess the compact attractor 𝒜 which is bounded in H2(Ω)×H2(Ω), where the nonlinear term f satisfies a critical exponential growth condition.


2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Jamel Benameur ◽  
Mongi Blel
Keyword(s):  

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.


2019 ◽  
Vol 20 (02) ◽  
pp. 2050013
Author(s):  
Alexandra Neamţu

We establish the existence of random stable and unstable manifolds for ill-posed stochastic partial differential equations (SPDEs). Namely, we assume that the linear part does not generate a [Formula: see text]-semigroup. Using the theory of integrated semigroups, we are able to analyze the long-time behavior of random dynamical systems generated by such SPDEs.


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