Initial Data Decay Research Articles

Local exponential stabilization for a class of Korteweg–de Vries equations by means of time-varying feedback laws

We study the exponential stabilization problem for a nonlinear Korteweg-de Vries equa- tion on bounded interval in cases where the linearized control system is not controllable. The system has Dirichlet boundary conditions at the end-points of the interval, a Neumann nonhomogeneous boundary condition at the right end-point which is the control. We build a class of time-varying feedback laws for which the solutions of the closed-loop systems with small initial data decay exponentially to 0. We present also results on the well-posedness of the closed-loop systems for general time-varying feedback laws.

Blowup time of solutions for a small diffusive parabolic problem with exponential source

In this paper, we study the asymptotic behavior of blowup time for a small diffusive parabolic equation with exponential source. We prove that the blowup time of the solution converges to that of the corresponding ODE as the small diffusive approaches zero. Moreover, we show a more accurate estimate of the blowup time. Precisely, when the initial data decay near the maximum, we obtain the lower and upper bound estimates of the blowup time with a higher-order term of the peak of initial data.

Uniform stability for solutions of a structural acoustics PDE model with no added dissipative feedback

A rate of rational decay is obtained for smooth solutions of a PDE model, which has been used in the literature to describe structural acoustic flows. This structural acoustics model is composed of two distinct PDE systems: (i) a wave equation, to model the interior acoustic flow within the given cavity Ω and (ii) a structurally damped elastic equation, to describe time‐evolving displacements along the flexible portion Γ0 of the cavity walls. Moreover, the extent of damping in this elastic component is quantified by parameter η∈[0,1]. The coupling between these two distinct dynamics occurs across the boundary interface Γ0. Our main result is the derivation of uniform decay rates for classical solutions of this particular structural acoustic PDE, decay rates that are obtained without incorporating any additional boundary dissipative feedback mechanisms. In particular, in the case that full Kelvin–Voight damping is present in fourth‐order elastic dynamics, that is, the structural acoustics system as it appears in the literature, solutions that correspond to smooth initial data decay at a rate of . By way of deriving these stability results, necessary a priori inequalities for a certain static structural acoustics PDE model are generated here; these inequalities ultimately allow for an application of a recently derived resolvent criterion for rational decay. Copyright © 2016 John Wiley & Sons, Ltd.

Infinite propagation speed and asymptotic behavior for a two-component Degasperis–Procesi system

In this paper, we consider the Cauchy problem for a two-component Degasperis–Procesi system. To begin with, we prove that the first component of the corresponding solution with initial compact support has an infinite propagation speed in the sense that it instantaneously loses its compact support, while the second component maintains the property of being compactly supported throughout its evolution. What’s more, we investigate the asymptotic behaviors of the solution at infinity within its lifespan provided the initial data decay exponentially and algebraically, respectively.

Persistence properties of the solutions to a generalized two-component Camassa–Holm shallow water system

This paper is concerned with the Cauchy problem for a generalized two-component Camassa–Holm shallow water system. We prove that the solution will maintain the corresponding properties at infinity within its lifespan provided the initial data decay exponentially and algebraically, respectively.

Admissible boundary values for the defocusing nonlinear Schrödinger equation with asymptotically time-periodic data

We consider solutions of the defocusing nonlinear Schrödinger equation in the quarter plane whose Dirichlet boundary data approach a single exponential αeiωt as t→∞. In order to determine the long time asymptotics of the solution, it is necessary to first characterize the asymptotic behavior of the Neumann value in terms of the given data. Assuming that the initial data decay as x→∞, we derive necessary conditions for the Neumann value to asymptote towards a single exponential of the form ceiωt. Since our approach yields expressions which relate α, ω, and c, the result can be viewed as a characterization of the large t behavior of the Dirichlet to Neumann map for single exponential profiles.

Global existence of timelike minimal surface of general co-dimension in Minkowski space time

In this paper, we prove that the global existence of solutions to timelike minimal surface equations having arbitrary co-dimension with slow decay initial data in two space dimensions and three space dimensions, provided that the initial value is suitably small.MSC:35L70.

On the moments of solutions to linear parabolic equations involving the biharmonic operator

We consider the solutions to Cauchy problems for the parabolic equation $u_\tau +\Delta^2u=0$ in $\mathbb{R}_+\times\mathbb{R}^n$, with fast decay initial data. We study the behavior of their moments. This enables us to give a more precise description of the sign-changing behavior of solutions corresponding to positive initial data.

Partial regularity of suitable weak solutions to the four‐dimensional incompressible magneto‐hydrodynamic equations

In this paper, we study the partial regularity of suitable weak solutions to the incompressible magneto‐hydrodynamic equations in dimension four by borrowing and improving the arguments given by Caffarelli, Kohn, and Nirenberg for incompressible Navier–Stokes equations. The so‐called ε‐regularity criteria are established for suitable weak solutions. As an application, an estimate on Hausdorff dimension of the possible singular points set for a suitable weak solution is given. Finally, we present further information on distribution of the possible singular points if the given initial data decay sufficiently rapidly or are not too singular at the origin, in some sense. Copyright © 2012 John Wiley & Sons, Ltd.

Breakdown for the Camassa–Holm Equation Using Decay Criteria and Persistence in Weighted Spaces

We exhibit a sufficient condition in terms of decay at infinity of the initial data for the finite time blowup of strong solutions to the Camassa–Holm equation: a wave breaking will occur as soon as the initial data decay faster at infinity than the solitons. In the case of data decaying slower than solitons we provide persistence results for the solution in weighted Lp-spaces for a large class of moderate weights. Explicit asymptotic profiles illustrate the optimality of these results.

Large time behavior of solutions to 3D compressible Navier-Stokes-Poisson system

We consider the three-dimensional compressible Navier-Stokes-Poisson system where the electric field of the internal electrostatic potential force is governed by the self-consistent Poisson equation. If the Fourier modes of the initial data are degenerate at the low frequency or the initial data decay fast at spatial infinity, we show that the density converges to its equilibrium state at the L 2-rate $$(1 + t)^{ - \tfrac{7} {4}}$$ or L ∞-rate $$(1 + t)^{ - \tfrac{5} {2}}$$ , and the momentum decays at the L 2-rate $$(1 + t)^{ - \tfrac{5} {4}}$$ or L ∞-rate (1 + t)−2. These convergence rates are shown to be optimal for the compressible Navier-Stokes-Poisson system.

The wave equation in a general spherically symmetric black hole geometry

We consider the Cauchy problem for the wave equation in a general class of spherically symmetric black hole geometries. Under certain mild conditions on the far-field decay and the singularity, we show that there is a unique globally smooth solution to the Cauchy problem for the wave equation with data compactly supported away from the horizon that is compactly supported for all times and decays in L 1 as t tends to infinity. We obtain as a corollary that in the geometry of black hole solutions of the SU(2) Einstein/Yang-Mills equations, solutions to the wave equation with compactly supported initial data decay as t goes to infinity.

Late-time tails of self-gravitating skyrmions

We consider the long-time behaviour of spherically symmetric solutions in the Einstein-Skyrme model. Using nonlinear perturbation analysis we obtain the leading order estimation of the tail in the topologically trivial sector (B = 0) of the model. We show that solutions starting from small compactly supported initial data decay as 1/t^4 at future timelike infinity and as 1/u^2 at future null infinity. We also verified that long-time behaviour for the tail in Einstein-Skyrme model is exactly the same as it was obtained for wave maps.

Life-Span of Classical Solutions to Hyperbolic Geometric Flow in Two Space Variables with Slow Decay Initial Data

In this paper we investigate the life-span of classical solutions to the hyperbolic geometric flow in two space variables with slow decay initial data. By establishing some new estimates on the solutions of linear wave equations in two space variables, we give a lower bound of the life-span of classical solutions to the hyperbolic geometric flow with asymptotic flat initial Riemann surfaces.

Dispersive stability of infinite-dimensional Hamiltonian systems on lattices

We derive dispersive stability results for oscillator chains like the FPU chain or the discrete Klein–Gordon chain. If the nonlinearity is of degree higher than four, then small localized initial data decay exactly as in the linear case. For this, we provide sharp decay estimates for the linearized problem using oscillatory integrals and avoiding the nonoptimal interpolation between different ℓ p spaces.

A proof of Price's Law on Schwarzschild black hole manifolds for all angular momenta

Price's Law states that linear perturbations of a Schwarzschild black hole fall off as t−2ℓ−3 for t→∞ provided the initial data decay sufficiently fast at spatial infinity. Moreover, if the perturbations are initially static (i.e., their time derivative is zero), then the decay is predicted to be t−2ℓ−4. We give a proof of t−2ℓ−2 decay for general data in the form of weighted L1 to L∞ bounds for solutions of the Regge–Wheeler equation. For initially static perturbations we obtain t−2ℓ−3. The proof is based on an integral representation of the solution which follows from self-adjoint spectral theory. We apply two different perturbative arguments in order to construct the corresponding spectral measure and the decay bounds are obtained by appropriate oscillatory integral estimates.

Note about late-time wave tails on a dynamical background

Consider a spherically symmetric spacetime generated by a self-gravitating massless scalar field $\phi$ and let $\psi$ be a test (nonspherical) massless scalar field propagating on this dynamical background. Gundlach, Price, and Pullin \cite{gpp2} computed numerically the late-time tails for different multipoles of the field $\psi$ and suggested that solutions with compactly supported initial data decay in accord with Price's law as $t^{-(2\ell+3)}$ at timelike infinity. We show that in the case of the time-dependent background Price's law holds only for $\ell=0$ while for each $\ell\geq 1$ the tail decays as $t^{-(2\ell+2)}$.

Point‐wise decay estimate for the global classical solutions to quasilinear hyperbolic systems

AbstractIn this paper, we first consider the Cauchy problem for quasilinear strictly hyperbolic systems with weak linear degeneracy. The existence of global classical solutions for small and decay initial data was established in (Commun. Partial Differential Equations 1994; 19:1263–1317; Nonlinear Anal. 1997; 28:1299–1322; Chin. Ann. Math. 2004; 25B:37–56). We give a new, very simple proof of this result and also give a sharp point‐wise decay estimate of the solution. Then, we consider the mixed initial‐boundary‐value problem for quasilinear hyperbolic systems with nonlinear boundary conditions in the first quadrant. Under the assumption that the positive eigenvalues are weakly linearly degenerate, the global existence of classical solution with small and decay initial and boundary data was established in (Discrete Continuous Dynamical Systems 2005; 12(1):59–78; Zhou and Yang, in press). We also give a simple proof of this result as well as a sharp point‐wise decay estimate of the solution. Copyright © 2008 John Wiley & Sons, Ltd.

Boundary controllability of Maxwell's equations with nonzero conductivity inside a cube, I: Spectral controllability

This is a first paper in a series of two. In both papers, we consider the question of control of Maxwell's equations in a homogeneous medium with positive conductivity by means of boundary surface currents. The domain under consideration is a cube, where the conductivity is allowed to take on any nonnegative value. An additional restriction imposed in order to make this problem more suitable for practical implementations is that the controls are applied over only one face of the cube. In this paper, the method of moments is employed to establish spectral controllability for the above case (meaning that any finite combination of eigenfunctions is controllable). In the companion paper [S.S. Krigman, C.E. Wayne, Boundary controllability of Maxwell's equations with nonzero conductivity inside a cube, II: Lack of exact controllability and controllability for very smooth solutions, J. Math. Anal. Appl. (2006), doi:10.1016/j.jmaa2006.02.102] it will be established, by modifying the calculations in [H.O. Fattorini, Estimates for sequences biorthogonal to certain complex exponentials and boundary control of the wave equation, in: New Trends in Systems Analysis, Proceedings of the International Symposium, Versailles, 1976, in: Lecture Notes in Control and Inform. Sci., vol. 2, Springer, Berlin, 1977, pp. 111–124], that exact controllability fails for this geometry regardless of the size of the conductivity term. However, we will also establish in [S.S. Krigman, C.E. Wayne, Boundary controllability of Maxwell's equations with nonzero conductivity inside a cube, II: Lack of exact controllability and controllability for very smooth solutions, J. Math. Anal. Appl. (2006), doi:10.1016/j.jmaa2006.02.102] controllability of solutions that are smooth enough that the Fourier coefficients of their initial data decay at a suitable exponential rate.

Boundary controllability of Maxwell's equations with nonzero conductivity inside a cube, II: Lack of exact controllability and controllability for very smooth solutions

This is a second paper in a two part series. In the prequel, [S.S. Krigman, C.E. Wayne, Boundary controllability of Maxwell's equations with nonzero conductivity inside a cube, I: Spectral controllability, J. Math. Anal. Appl. (2006), doi:10.1016/j.jmaa2006.06.101], we showed that a system of Maxwell's equations for a homogeneous medium in a cube with nonnegative conductivity possesses the property that any finite combination of eigenfunctions is controllable (spectral controllability) by means of boundary surface currents applied over only one face of the cube. In the present paper it is established, by modifying the calculations in [H.O. Fattorini, Estimates for sequences biorthogonal to certain complex exponentials and boundary control of the wave equation, in: New Trends in Systems Analysis, Proceedings of the International Symposium, Versailles, 1976, in: Lecture Notes in Control and Inform. Sci., vol. 2, Springer, Berlin, 1977, pp. 111–124], that spectral controllability is the strongest result possible for this geometry, since the exact controllability fails regardless of the size of the conductivity term. However, we do establish controllability of solutions that are smooth enough that the Fourier coefficients of their initial data decay at an appropriate exponential rate. This does not contradict the lack of exact controllability since in any Sobolev space there are initial conditions which violate these restrictions.