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Kenji Nishihara - One of the best experts on this subject based on the ideXlab platform.
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decay properties for the Damped Wave Equation with space dependent potential and absorbed semilinear term
Communications in Partial Differential Equations, 2010Co-Authors: Kenji NishiharaAbstract:We consider the Cauchy problem for the Damped Wave Equation with space dependent potential V(x)u t and absorbed semilinear term |u|ρ−1 u in R N . Our assumption on V(x) ∼ (1 + |x|2)−α/2 (0 ≤ α <1) ...
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decay properties for the Damped Wave Equation with space dependent potential and absorbed semilinear term
Communications in Partial Differential Equations, 2010Co-Authors: Kenji NishiharaAbstract:We consider the Cauchy problem for the Damped Wave Equation with space dependent potential V(x)u t and absorbed semilinear term |u|ρ−1 u in R N . Our assumption on V(x) ∼ (1 + |x|2)−α/2 (0 ≤ α ρ c (N, α): = 1 + 2/(N − α) and for ρ < ρ c (N, α). We believe that in the “supercritical” exponent the decay rates obtained are almost the same as those for the linear parabolic problem, while, in the “subcritical” exponent the solution decays faster than that of linear Equation, thanks to the absorbed semilinear term. So we believe that ρ c (N, α) is a critical exponent. Note that ρ c (N, α) with α = 0 coincides to the Fujita exponent .
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l2 estimates of solutions for Damped Wave Equations with space time dependent damping term
Journal of Differential Equations, 2010Co-Authors: Jiayun Lin, Kenji Nishihara, Jian ZhaiAbstract:Abstract In this paper, we consider the Damped Wave Equation with space–time dependent potential b ( t , x ) and absorbing semilinear term | u | ρ − 1 u . Here, b ( t , x ) = b 0 ( 1 + | x | 2 ) − α 2 ( 1 + t ) − β with b 0 > 0 , α , β ⩾ 0 and α + β ∈ [ 0 , 1 ) . Based on the local existence theorem, we obtain the global existence and the L 2 decay rate of the solution by using the weighted energy method. The decay rate coincides with the result of Nishihara [K. Nishihara, Decay properties for the Damped Wave Equation with space dependent potential and absorbed semilinear term, preprint] in the case of β = 0 and coincides with the result of Nishihara and Zhai [K. Nishihara, J. Zhai, Asymptotic behaviors of time dependent Damped Wave Equations, preprint] in the case of α = 0 .
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asymptotic behavior of solutions for the Damped Wave Equation with slowly decaying data
Journal of Mathematical Analysis and Applications, 2008Co-Authors: Takashi Narazaki, Kenji NishiharaAbstract:Abstract We consider the Cauchy problem for the Damped Wave Equation u t t − Δ u + u t = | u | ρ − 1 u , ( t , x ) ∈ R + × R N and the heat Equation ϕ t − Δ ϕ = | ϕ | ρ − 1 ϕ , ( t , x ) ∈ R + × R N . If the data is small and slowly decays likely c 1 ( 1 + | x | ) − k N , 0 k ⩽ 1 , then the critical exponent is ρ c ( k ) = 1 + 2 k N for the semilinear heat Equation. In this paper it is shown that in the supercritical case there exists a unique time global solution to the Cauchy problem for the semilinear heat Equation in any dimensional space R N , whose asymptotic profile is given by Φ 0 ( t , x ) = ∫ R N e − | x − y | 2 4 t ( 4 π t ) N / 2 c 1 ( 1 + | y | 2 ) k N / 2 d y provided that the data ϕ 0 satisfies lim | x | → ∞ 〈 x 〉 k N ϕ 0 ( x ) = c 1 ( ≠ 0 ) . Even in the semilinear Damped Wave Equation in the supercritical case a time global solution u with the data ( u , u t ) ( 0 , x ) = ( u 0 , u 1 ) ( x ) is shown in low dimensional spaces R N , N = 1 , 2 , 3 , to have the same asymptotic profile Φ 0 ( t , x ) provided that lim | x | → ∞ 〈 x 〉 k N ( u 0 + u 1 ) ( x ) = c 1 ( ≠ 0 ) . Those proofs are given by elementary estimates on the explicit formulas of solutions.
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global asymptotics for the Damped Wave Equation with absorption in higher dimensional space
Journal of The Mathematical Society of Japan, 2006Co-Authors: Kenji NishiharaAbstract:We consider the Cauchy problem for the Damped Wave Equation with absorption u t t - Δ u + u t + | u | ρ - 1 u = 0 , ( t , x ) ∈ R + × R N , ( * ) with N = 3 , 4 . The behavior of u as t → ∞ is expected to be the Gauss kernel in the supercritical case ρ > ρ c ( N ) : = 1 + 2 / N . In fact, this has been shown by Karch [12] (Studia Math., 143 (2000), 175--197) for ρ > 1 + 4 N ( N = 1 , 2 , 3 ) , Hayashi, Kaikina and Naumkin [8] (preprint (2004)) for ρ > ρ c ( N ) ( N = 1 ) and by Ikehata, Nishihara and Zhao [11] (J. Math. Anal. Appl., 313 (2006), 598--610) for ρ c ( N ) < ρ ≤ 1 + 4 N ( N = 1 , 2 ) and ρ c ( N ) < ρ < 1 + 3 N ( N = 3 ) . Developing their result, we will show the behavior of solutions for ρ c ( N ) < ρ ≤ 1 + 4 N ( N = 3 ) , ρ c ( N ) < ρ < 1 + 4 N ( N = 4 ) . For the proof, both the weighted L 2 -energy method with an improved weight developed in Todorova and Yordanov [22] (J. Differential Equations, 174 (2001), 464--489) and the explicit formula of solutions are still usefully used. This method seems to be not applicable for N = 5 , because the semilinear term is not in C 2 and the second derivatives are necessary when the explicit formula of solutions is estimated.
Nicolas Burq - One of the best experts on this subject based on the ideXlab platform.
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decay for the kelvin voigt Damped Wave Equation piecewise smooth damping
arXiv: Analysis of PDEs, 2020Co-Authors: Nicolas Burq, Chenmin SunAbstract:We study the energy decay rate of the Kelvin-Voigt Damped Wave Equation with piecewise smooth damping on the multi-dimensional domain. Under suitable geometric assumptions on the support of the damping, we obtain the optimal polynomial decay rate which turns out to be different from the one-dimensional case studied in [LR05]. This optimal decay rate is saturated by high energy quasi-modes localised on geometric optics rays which hit the interface along non orthogonal neither tangential directions. The proof uses semi-classical analysis of boundary value problems.
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Laplace Eigenfunctions And Damped Wave Equation Ii: Product Manifolds
arXiv: Analysis of PDEs, 2015Co-Authors: Nicolas Burq, Claude ZuilyAbstract:- The purpose of this article is to study possible concentrations of eigenfunc-tions of Laplace operators (or more generally quasi-modes) on product manifolds. We show that the approach of the first author and Zworski [10, 11] applies (modulo rescalling) and deduce new stabilization results for weakly Damped Wave Equations which extend to product manifolds previous results by Leautaud-Lerner [12] obtained for products of tori.
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imperfect geometric control and overdamping for the Damped Wave Equation
Communications in Mathematical Physics, 2015Co-Authors: Nicolas Burq, Hans ChristiansonAbstract:We consider the Damped Wave Equation on a manifold with imperfect geometric control. We show the sub-exponential energy decay estimate in (Christianson, J Funct Anal 258(3):1060–1065, 2010) is optimal in the case of one hyperbolic periodic geodesic. We show if the Equation is overDamped, then the energy decays exponentially. Finally we show if the Equation is overDamped but geometric control fails for one hyperbolic periodic geodesic, then nevertheless the energy decays exponentially.
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exponential decay for the Damped Wave Equation in unbounded domains
arXiv: Analysis of PDEs, 2014Co-Authors: Nicolas Burq, Romain JolyAbstract:We study the decay of the semigroup generated by the Damped Wave Equation in an unbounded domain. We first prove under the natural geometric control condition the exponential decay of the semigroup. Then we prove under a weaker condition the logarithmic decay of the solutions (assuming that the initial data are smoother). As corollaries, we obtain several extensions of previous results of stabilisation and control.
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imperfect geometric control and overdamping for the Damped Wave Equation
arXiv: Analysis of PDEs, 2013Co-Authors: Nicolas Burq, Hans ChristiansonAbstract:We consider the Damped Wave Equation on a manifold with imperfect geometric control. We show the sub-exponential energy decay estimate in \cite{Chr-NC-erratum} is optimal in the case of one hyperbolic periodic geodesic. We show if the Equation is overDamped, then the energy decays exponentially. Finally we show if the Equation is overDamped but geometric control fails for one hyperbolic periodic geodesic, then nevertheless the energy decays exponentially.
Yuta Wakasugi - One of the best experts on this subject based on the ideXlab platform.
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unconditional well posedness for the energy critical nonlinear Damped Wave Equation
Journal of Evolution Equations, 2021Co-Authors: Takahisa Inui, Yuta WakasugiAbstract:For the energy-critical nonlinear Damped Wave Equation, we show the unconditional well-posedness. The unconditional well-posedness means local well-posedness and the unconditional uniqueness. First, we give the local well-posedness and stability, whose statement will be useful to investigate the global dynamics. At second, we show the unconditional uniqueness. Since these problems seem not to be solvable as a direct perturbation of the Wave Equation, we apply the Strichartz estimates for the Damped Wave Equation including the Wave endpoint case.
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l p l q estimates for the Damped Wave Equation and the critical exponent for the nonlinear problem with slowly decaying data
Communications on Pure and Applied Analysis, 2019Co-Authors: Masahiro Ikeda, Takahisa Inui, Mamoru Okamoto, Yuta WakasugiAbstract:We study the Cauchy problem of the Damped Wave Equation \begin{document}$ \begin{align*} \partial_{t}^2 u - \Delta u + \partial_t u = 0 \end{align*} $\end{document} and give sharp \begin{document}$ L^p $\end{document} - \begin{document}$ L^q $\end{document} estimates of the solution for \begin{document}$ 1\le q \le p with derivative loss. This is an improvement of the so-called Matsumura estimates. Moreover, as its application, we consider the nonlinear problem with initial data in \begin{document}$ (H^s\cap H_r^{\beta}) \times (H^{s-1} \cap L^r) $\end{document} with \begin{document}$ r \in (1,2] $\end{document} , \begin{document}$ s\ge 0 $\end{document} , and \begin{document}$ \beta = (n-1)|\frac{1}{2}-\frac{1}{r}| $\end{document} , and prove the local and global existence of solutions. In particular, we prove the existence of the global solution with small initial data for the critical nonlinearity with the power \begin{document}$ 1+\frac{2r}{n} $\end{document} , while it is known that the critical power \begin{document}$ 1+\frac{2}{n} $\end{document} belongs to the blow-up region when \begin{document}$ r = 1 $\end{document} . We also discuss the asymptotic behavior of the global solution in supercritical cases. Moreover, we present blow-up results in subcritical cases. We give estimates of lifespan by an ODE argument.
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l p l q estimates for the Damped Wave Equation and the critical exponent for the nonlinear problem with slowly decaying data
arXiv: Analysis of PDEs, 2017Co-Authors: Masahiro Ikeda, Takahisa Inui, Mamoru Okamoto, Yuta WakasugiAbstract:We study the Cauchy problem of the Damped Wave Equation \begin{align*} \partial_{t}^2 u - \Delta u + \partial_t u = 0 \end{align*} and give sharp $L^p$-$L^q$ estimates of the solution for $1\le q \le p < \infty\ (p\neq 1)$ with derivative loss. This is an improvement of the so-called Matsumura estimates. Moreover, as its application, we consider the nonlinear problem with initial data in $(H^s\cap H_r^{\beta}) \times (H^{s-1} \cap L^r)$ with $r \in (1,2]$, $s\ge 0$, and $\beta = (n-1)|\frac{1}{2}-\frac{1}{r}|$, and prove the local and global existence of solutions. In particular, we prove the existence of the global solution with small initial data for the critical nonlinearity with the power $1+\frac{2r}{n}$, while it is known that the critical power $1+\frac{2}{n}$ belongs to the blow-up region when $r=1$. We also discuss the asymptotic behavior of the global solution in supercritical cases. Moreover, we present blow-up results in subcritical cases. We give estimates of lifespan and blow-up rates by an ODE argument.
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scaling variables and asymptotic profiles for the semilinear Damped Wave Equation with variable coefficients
Journal of Mathematical Analysis and Applications, 2017Co-Authors: Yuta WakasugiAbstract:Abstract We study the asymptotic behavior of solutions for the semilinear Damped Wave Equation with variable coefficients. We prove that if the damping is effective, and the nonlinearity and other lower order terms can be regarded as perturbations, then the solution is approximated by the scaled Gaussian of the corresponding linear parabolic problem. The proof is based on the scaling variables and energy estimates.
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The Cauchy problem for the nonlinear Damped Wave Equation with slowly decaying data
Nonlinear Differential Equations and Applications NoDEA, 2017Co-Authors: Masahiro Ikeda, Takahisa Inui, Yuta WakasugiAbstract:We study the Cauchy problem for the nonlinear Damped Wave Equation and establish the large data local well-posedness and small data global well-posedness with slowly decaying initial data. We also prove that the asymptotic profile of the global solution is given by a solution of the corresponding parabolic problem, which shows that the solution of the Damped Wave Equation has the diffusion phenomena. Moreover, we show blow-up of solution and give the estimate of the lifespan for a subcritical nonlinearity. In particular, we determine the critical exponent for any space dimension.
Takahisa Inui - One of the best experts on this subject based on the ideXlab platform.
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unconditional well posedness for the energy critical nonlinear Damped Wave Equation
Journal of Evolution Equations, 2021Co-Authors: Takahisa Inui, Yuta WakasugiAbstract:For the energy-critical nonlinear Damped Wave Equation, we show the unconditional well-posedness. The unconditional well-posedness means local well-posedness and the unconditional uniqueness. First, we give the local well-posedness and stability, whose statement will be useful to investigate the global dynamics. At second, we show the unconditional uniqueness. Since these problems seem not to be solvable as a direct perturbation of the Wave Equation, we apply the Strichartz estimates for the Damped Wave Equation including the Wave endpoint case.
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remark on asymptotic order for the energy critical nonlinear Damped Wave Equation to the linear heat Equation via the strichartz estimates
2020Co-Authors: Takahisa InuiAbstract:We consider the Damped Wave Equation with the energy critical power type nonlinearity. It is known that the global solution with a finite space-time norm decays to 0 as time goes to infinity. In the present paper, we give the asymptotic order that the solution goes to a solution of the linear heat Equation.
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the strichartz estimates for the Damped Wave Equation and the behavior of solutions for the energy critical nonlinear Equation
Nodea-nonlinear Differential Equations and Applications, 2019Co-Authors: Takahisa InuiAbstract:For the linear Damped Wave Equation (DW), the $$L^p$$–$$L^q$$ type estimates have been well studied. Recently, Watanabe (RIMS Kokyuroku Bessatsu B 63:77–101, 2017) showed the Strichartz estimates for DW when $$d=2,3$$. In the present paper, we give Strichartz estimates for DW in higher dimensions. Moreover, by applying the estimates, we give the local well-posedness of the energy critical nonlinear Damped Wave Equation (NLDW) $$\partial _t^2 u - \Delta u +\partial _t u = |u|^{\frac{4}{d-2}}u$$, $$(t,x) \in [0,T) \times {\mathbb {R}}^d$$, where $$3 \le d \le 5$$. Especially, we show the small data global existence for NLDW. In addition, we investigate the behavior of the solutions to NLDW. Namely, we give a decay result for solutions with finite Strichartz norm and a blow-up result for solutions with negative Nehari functional.
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the strichartz estimates for the Damped Wave Equation and the behavior of solutions for the energy critical nonlinear Equation
arXiv: Analysis of PDEs, 2019Co-Authors: Takahisa InuiAbstract:For the linear Damped Wave Equation (DW), the $L^p$-$L^q$ type estimates have been well studied. Recently, Watanabe showed the Strichartz estimates for DW when $d=2,3$. In the present paper, we give Strichartz estimates for DW in higher dimensions. Moreover, by applying the estimates, we give the local well-posedness of the energy critical nonlinear Damped Wave Equation (NLDW) $\partial_t^2 u - \Delta u +\partial_t u = |u|^{\frac{4}{d-2}}u$, $(t,x) \in [0,T) \times \mathbb{R}^d$, where $3 \leq d \leq 5$. Especially, we show the small data global existence for NLDW. In addition, we investigate the behavior of the solutions to NLDW. Namely, we give a decay result for solutions with finite Strichartz norm and a blow-up result for solutions with negative Nehari functional.
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l p l q estimates for the Damped Wave Equation and the critical exponent for the nonlinear problem with slowly decaying data
Communications on Pure and Applied Analysis, 2019Co-Authors: Masahiro Ikeda, Takahisa Inui, Mamoru Okamoto, Yuta WakasugiAbstract:We study the Cauchy problem of the Damped Wave Equation \begin{document}$ \begin{align*} \partial_{t}^2 u - \Delta u + \partial_t u = 0 \end{align*} $\end{document} and give sharp \begin{document}$ L^p $\end{document} - \begin{document}$ L^q $\end{document} estimates of the solution for \begin{document}$ 1\le q \le p with derivative loss. This is an improvement of the so-called Matsumura estimates. Moreover, as its application, we consider the nonlinear problem with initial data in \begin{document}$ (H^s\cap H_r^{\beta}) \times (H^{s-1} \cap L^r) $\end{document} with \begin{document}$ r \in (1,2] $\end{document} , \begin{document}$ s\ge 0 $\end{document} , and \begin{document}$ \beta = (n-1)|\frac{1}{2}-\frac{1}{r}| $\end{document} , and prove the local and global existence of solutions. In particular, we prove the existence of the global solution with small initial data for the critical nonlinearity with the power \begin{document}$ 1+\frac{2r}{n} $\end{document} , while it is known that the critical power \begin{document}$ 1+\frac{2}{n} $\end{document} belongs to the blow-up region when \begin{document}$ r = 1 $\end{document} . We also discuss the asymptotic behavior of the global solution in supercritical cases. Moreover, we present blow-up results in subcritical cases. We give estimates of lifespan by an ODE argument.
Anna Persson - One of the best experts on this subject based on the ideXlab platform.
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a generalized finite element method for the strongly Damped Wave Equation with rapidly varying data
Mathematical Modelling and Numerical Analysis, 2021Co-Authors: Per Ljung, Axel Malqvist, Anna PerssonAbstract:We propose a generalized finite element method for the strongly Damped Wave Equation with highly varying coefficients. The proposed method is based on the localized orthogonal decomposition introduced in Malqvist and Peterseim [Math. Comp. 83 (2014) 2583–2603], and is designed to handle independent variations in both the damping and the Wave propagation speed respectively. The method does so by automatically correcting for the damping in the transient phase and for the propagation speed in the steady state phase. Convergence of optimal order is proven in L2 (H1 )-norm, independent of the derivatives of the coefficients. We present numerical examples that confirm the theoretical findings.
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a generalized finite element method for the strongly Damped Wave Equation with rapidly varying data
arXiv: Numerical Analysis, 2020Co-Authors: Per Ljung, Axel Malqvist, Anna PerssonAbstract:We propose a generalized finite element method for the strongly Damped Wave Equation with highly varying coefficients. The proposed method is based on the localized orthogonal decomposition introduced and is designed to handle independent variations in both the damping and the Wave propagation speed respectively. The method does so by automatically correcting for the damping in the transient phase and for the propagation speed in the steady state phase. Convergence of optimal order is proven in $L_2(H^1)$-norm, independent of the derivatives of the coefficients. We present numerical examples that confirm the theoretical findings.
