The energy decay problem for wave equations with nonlinear dissipative terms in $\mathbb{R}^n$

2007 ◽  
Vol 56 (1) ◽  
pp. 389-416 ◽  
Author(s):  
Grozdena Todorova ◽  
Borislav Yordanov
Keyword(s):  
Wave Equations ◽  
Energy Decay ◽  
Dissipative Terms

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 Energy decay of vortices in viscous fluids: an applied mathematics view

2012 ◽  
Vol 709 ◽  
pp. 593-609 ◽  
Author(s):  
Jan Nordström ◽  
Björn Lönn
Keyword(s):  
Numerical Solutions ◽  
Stokes Equations ◽  
Applied Mathematics ◽  
Energy Decay ◽  
Viscous Fluids ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
The Matrix ◽  
High Order Finite Difference ◽  
Dissipative Terms

AbstractThe energy decay of vortices in viscous fluids governed by the compressible Navier–Stokes equations is investigated. It is shown that the main reason for the slow decay is that zero eigenvalues exist in the matrix related to the dissipative terms. The theoretical analysis is purely mathematical and based on the energy method. To check the validity of the theoretical result in practice, numerical solutions to the Navier–Stokes equations are computed using a stable high-order finite difference method. The numerical computations corroborate the theoretical conclusion.

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