Decay estimates of solutions of a nonlinearly damped semilinear wave equation

Aissa Guesmia; Salim A. Messaoudi

Displaying similar documents to “Decay estimates of solutions of a nonlinearly damped semilinear wave equation”

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Let $P$ be a long range metric perturbation of the Euclidean Laplacian on $ℝ^{d}$ , $d \geq 2$ . We prove local energy decay for the solutions of the wave, Klein-Gordon and Schrödinger equations associated to $P$ . The problem is decomposed in a low and high frequency analysis. For the high energy part, we assume a non trapping condition. For low (resp. high) frequencies we obtain a general result about the local energy decay for the group $e^{i t f (P)}$ where $f$ has a suitable development at zero (resp. infinity). ...

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The present paper is particularly devoted to the damping effect in ferromagnetic materials. We are interested in determining the sensitivity of the LLG method solution to the phenomenological damping parameter a. We discuss the behaviour of the global weak solutions with finite energy of the Landau-Lifshitz equations when the damping parameter a tends either to 0 (underdamped case) or $+\infty$ (overdamped case).

Large data local solutions for the derivative NLS equation

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Selfsimilar profiles in large time asymptotics of solutions to damped wave equations

Grzegorz Karch (2000)

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Large time behavior of solutions to the generalized damped wave equation $u_{t t} + A u_{t} + ν B u + F (x, t, u, u_{t}, \nabla u) = 0$ for $(x, t) \in ℝ^{n} \times [0, \infty)$ is studied. First, we consider the linear nonhomogeneous equation, i.e. with F = F(x,t) independent of u. We impose conditions on the operators A and B, on F, as well as on the initial data which lead to the selfsimilar large time asymptotics of solutions. Next, this abstract result is applied to the equation where $A u_{t} = u_{t}$ , $B u = - Δ u$ , and the nonlinear term is either $| u_{t} |^{q - 1} u_{t}$ or ${| u |}^{α - 1} u$ . In this case, the asymptotic profile of solutions...

Dispersive and Strichartz estimates on H-type groups

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Our purpose is to generalize the dispersive inequalities for the wave equation on the Heisenberg group, obtained in [1], to H-type groups. On those groups we get optimal time decay for solutions to the wave equation (decay as $t^{- p / 2}$ ) and the Schrödinger equation (decay as $t^{(1 - p) / 2}$ ), p being the dimension of the center of the group. As a corollary, we obtain the corresponding Strichartz inequalities for the wave equation, and, assuming that p > 1, for the Schrödinger equation.

Global well-posedness for the Klein-Gordon-Schrödinger system with higher order coupling

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Global well-posedness for the Klein-Gordon-Schrödinger system with generalized higher order coupling, which is a system of PDEs in two variables arising from quantum physics, is proven. It is shown that the system is globally well-posed in $(u, n) \in L^{2} \times L^{2}$ under some conditions on the nonlinearity (the coupling term), by using the $L^{2}$ conservation law for $u$ and controlling the growth of $n$ via the estimates in the local theory. In particular, this extends the well-posedness results for such a system in...

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Existence and decay in non linear viscoelasticity

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Bollettino dell'Unione Matematica Italiana

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In this work we study the existence, uniqueness and decay of solutions to a class of viscoelastic equations in a separable Hilbert space $H$ given by $\begin{matrix} u_{t t} + M ([u]) A u - \int_{0}^{t} g (t - τ) N ([u]) A u d τ = 0, in L^{2} (0, T; H) \\ u (0) = u_{0}, u_{t} (0) = u_{1} \end{matrix}$ where by $[u (t)]$ we are denoting $[u (t)] = ((u (t), u_{t} (t), (A u (t), u_{t} (t)), ∥ A^{\frac{1}{2}} u (t) ∥^{2}, ∥ A^{\frac{1}{2}} u_{t} (t) ∥^{2}, ∥ A u (t) ∥^{2}) \in ℝ^{5}$ $A : D (A) \subset H \to H$ is a nonnegative, self-adjoint operator, $M$ , $N : R^{5} \to R$ are $C^{2}$ - functions and $g : R \to R$ is a $C^{3}$ -function with appropriates conditions. We show that there exists global solution in time for small initial data. When $[u (t)] = {∥A^{\frac{1}{2}} u∥}^{2}$ and $N = 1$ , we show the global existence for large initial data $(u_{0}, u_{1})$ taken in the space $D (A) D (A^{1 / 2})$ provided they are...

Global regularity for the 3D MHD system with damping

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