Lp − Lq Decay Estimates for Wave Equations with Time-Dependent Coefficients

2004 ◽  
Vol 11 (4) ◽  
pp. 534-548 ◽  
Author(s):  
Michael Reissig
Keyword(s):  
Wave Equations ◽  
Time Dependent ◽  
Decay Estimates

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.

scholarly journals Scattering for critical wave equations with variable coefficients

2021 ◽  
pp. 1-19
Author(s):  
Shi-Zhuo Looi ◽  
Mihai Tohaneanu
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Variable Coefficients ◽  
Energy Decay ◽  
Time Dependent ◽  
Linear Wave ◽  
Decay Estimates ◽  
Local Energy ◽  
Linear Wave Equation ◽  
Semilinear Wave Equation

Abstract We prove that solutions to the quintic semilinear wave equation with variable coefficients in ${{\mathbb {R}}}^{1+3}$ scatter to a solution to the corresponding linear wave equation. The coefficients are small and decay as $|x|\to \infty$ , but are allowed to be time dependent. The proof uses local energy decay estimates to establish the decay of the $L^{6}$ norm of the solution as $t\to \infty$ .

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 Existence and decay of finite energy solutions for semilinear dissipative wave equations in time-dependent domains

2020 ◽  
Vol 40 (6) ◽  
pp. 725-736
Author(s):  
Mitsuhiro Nakao
Keyword(s):  
Existence And Uniqueness ◽  
Wave Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Finite Energy ◽  
Time Dependent ◽  
Energy Solution ◽  
Noncylindrical Domain ◽  
Finite Energy Solution ◽  
Initial Boundary Value

We consider the initial-boundary value problem for semilinear dissipative wave equations in noncylindrical domain \(\bigcup_{0\leq t \lt\infty} \Omega(t)\times\{t\} \subset \mathbb{R}^N\times \mathbb{R}\). We are interested in finite energy solution. We derive an exponential decay of the energy in the case \(\Omega(t)\) is bounded in \(\mathbb{R}^N\) and the estimate \[\int\limits_0^{\infty} E(t)dt \leq C(E(0),\|u(0)\|)< \infty\] in the case \(\Omega(t)\) is unbounded. Existence and uniqueness of finite energy solution are also proved.

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