On exponential stability of a semilinear wave equation with variable coefficients under the nonlinear boundary feedback

The uniform stabilization of an originally regarded nondissipative system described by a semilinear wave equation with variable coefficients under the nonlinear boundary feedback is considered. The existence of both weak and strong solutions to the system is proven by the Galerkin method. The exponential stability of the system is obtained by introducing an equivalent energy function and using the energy multiplier method on the Riemannian manifold. This equivalent energy function shows particularly that the system is essentially a dissipative system. This result not only generalizes the result from constant coefficients to variable coefficients for these kinds of semilinear wave equations but also simplifies significantly the proof for constant coefficients case considered in [A. Guesmia, A new approach of stabilization of nondissipative distributed systems, SIAM J. Control Optim. 42 (2003) 24–52] where the system is claimed to be nondissipative.     

Local uniform stability for the semilinear wave equation in inhomogeneous media with locally distributed Kelvin–Voigt damping

We consider the semilinear wave equation posed in an inhomogeneous medium Ω with smooth boundary subject to a local viscoelastic damping distributed around a neighborhood ω of the boundary according to the Geometric Control Condition. We show that the energy of the wave equation goes uniformly and exponentially to zero for all initial data of finite energy taken in bounded sets of finite energy phase‐space. As far as we know, this is the first stabilization result for a semilinear wave equation with localized Kelvin–Voigt damping.     

On Boundary Stability of Wave Equations with Variable Coefficients

In this paper, we consider the boundary stabilization of the wave equation with variable coefficients by Riemmannian geometry method subject to a different geometric condition which is motivated by the geometric multiplier identities. Several (multiplier) identities (inequalities) which have been built for constant wave equation by Kormornik and Zuazua are generalized to the variable coefficient case by some computational techniques in Riemmannian geometry, so that the precise estimates on the exponential decay rate are derived from those inequalitities. Also, the exponential decay for the solutions of semilinear wave equation with variable coefficients is obtained under natural growth and sign assumptions on the nonlinearity. Our method is rather general and can be adapted to other evolution systems with variable coefficients (e.g. elasticity plates) as well.     

Remarks on the energy decay for nonlinear wave equations with nonlinear localized dissipative terms

We derive an energy decay estimate for solutions to the initial-boundary value problem of a semilinear wave equation with a nonlinear localized dissipation. To overcome a difficulty related to derivative-loss mechanism we employ a ‘loan’ method.     

Stabilization of a transmission wave/plate equation

We consider a stabilization problem, for a model arising in the control of noise, coupling the damped wave equation with a damped Kirchhoff plate equation. We prove an exponential stability result under some geometric condition. Our method is based on an identity with multipliers that allows to show an appropriate energy estimate.     

Gain of analyticity for semilinear Schrödinger equations

We discuss gain of analyticity phenomenon of solutions to the initial value problem for semilinear Schrödinger equations with gauge invariant nonlinearity. We prove that if the initial data decays exponentially, then the solution becomes real-analytic in the space variable and a Gevrey function of order 2 in the time variable except in the initial plane. Our proof is based on the energy estimates developed in our previous work and on fine summation formulae concerned with a matrix norm.     

Decay Rates of Interactive Hyperbolic-Parabolic PDE Models with Thermal Effects on the Interface

We consider coupled PDE systems comprising of a hyperbolic and a parabolic-like equation with an interface on a portion of the boundary. These models are motivated by structural acoustic problems. A specific prototype consists of a wave equation defined on a three-dimensional bounded domain Ω coupled with a thermoelastic plate equation defined on Γ ₀—a flat surface of the boundary \partial Ω . Thus, the coupling between the wave and the plate takes place on the interface Γ ₀. The main issue studied here is that of uniform stability of the overall interactive model. Since the original (uncontrolled) model is only strongly stable, but not uniformly stable, the question becomes: what is the ``minimal amount' of dissipation necessary to obtain uniform decay rates for the energy of the overall system? Our main result states that boundary nonlinear dissipation placed only on a suitable portion of the part of the boundary which is complementary to Γ <sub>0</sub>, suffices for the stabilization of the entire structure. This result is new with respect to the literature on several accounts: (i) thermoelasticity is accounted for in the plate model; (ii) the plate model does not account for any type of mechanical damping, including the structural damping most often considered in the literature; (iii) there is no mechanical damping placed on the interface Γ <sub>0</sub>; (iv) the boundary damping is nonlinear without a prescribed growth rate at the origin; (v) the undamped portions of the boundary \partial Ω are subject to Neumann (rather than Dirichlet) boundary conditions, which is a recognized difficulty in the context of stabilization of wave equations, due to the fact that the strong Lopatinski condition does not hold. The main mathematical challenge is to show how the thermal energy is propagated onto the hyperbolic component of the structure. This is achieved by using a recently developed sharp theory of boundary traces corresponding to wave and plate equations, along with the analytic estimates recently established for the co-continuous semigroup associated with thermal plates subject to free boundary conditions. These trace inequalities along with the analyticity of the thermoelastic plate component allow one to establish appropriate inverse/ recovery type estimates which are critical for uniform stabilization. Our main result provides ``optimal' uniform decay rates for the energy function corresponding to the full structure. These rates are described by a suitable nonlinear ordinary differential equation, whose coefficients depend on the growth of the nonlinear dissipation at the origin. \par Accepted 12 May 2000. Online publication 6 October 2000.     

A note on geometric conditions for boundary control of wave equations with variable coefficients

The analytical condition given by Wyler for boundary stabilization of wave equations with variable coefficients is compared with the geometrical condition derived by Yao in terms of the Riemannian geometry method for exact controllability of wave equations with variable coefficients. It is shown that these two conditions are equivalent.     

A Gamma-convergence argument for the blow-up of a non-local semilinear parabolic equation with Neumann boundary conditions

In this paper we study a simple non-local semilinear parabolic equation in a bounded domain with Neumann boundary conditions. We obtain a global existence result for initial data whose L^∞$L^{\infty }$-norm is less than a constant depending explicitly on the geometry of the domain. A natural energy is associated to the equation and we establish a relationship between the finite-time blow up of solutions and the negativity of their energy. The proof of this result is based on a Gamma-convergence technique.     

Global conservative solutions for a model equation for shallow water waves of moderate amplitude

In this paper, we study the continuation of solutions to an equation for surface water waves of moderate amplitude in the shallow water regime beyond wave breaking (in [11], Constantin and Lannes proved that this equation accommodates wave breaking phenomena). Our approach is based on a method proposed by Bressan and Constantin [2]. By introducing a new set of independent and dependent variables, which resolve all singularities due to possible wave breaking, the evolution problem is rewritten as a semilinear system. Local existence of the semilinear system is obtained as fixed points of a contractive transformation. Moreover, this formulation allows one to continue the solution after collision time, giving a global conservative solution where the energy is conserved for almost all times. Finally, returning to the original variables, we obtain a semigroup of global conservative solutions, which depend continuously on the initial data.     

Remarks on a weighted energy estimate and its application to nonlinear wave equations in one space dimension

A weighted energy estimate with tangential derivatives on the light cone is applied for the Cauchy problem of semilinear wave equations with the null conditions in one space dimension. The well-posedness and lifespan of the solutions are considered based on the vector field method.     

On the asymptotic behavior of the energy for the wave equations with time depending coefficients

We consider the asymptotic behavior of the total energy of solutions to the Cauchy problem for wave equations with time dependent propagation speed. The main purpose of this paper is that the asymptotic behavior of the total energy is dominated by the following properties of the coefficient: order of the differentiability, behavior of the derivatives as t → ∞ and stabilization of the amplitude described by an integral. Moreover, the optimality of these properties are ensured by actual examples. Supported by Grants-in-Aid for Young Scientists (B) (No.16740098), The Ministry of Education, Culture, Sports, Science and Technology.     

Stabilization of a one‐dimensional wave equation with variable coefficient under non‐collocated control and delayed observation

In this paper, we consider stabilization of a 1‐dimensional wave equation with variable coefficient where non‐collocated boundary observation suffers from an arbitrary time delay. Since input and output are non‐collocated with each other, it is more complex to design the observer system. After showing well‐posedness of the open‐loop system, the observer and predictor systems are constructed to give the estimated state feedback controller. Different from the partial differential equation with constant coefficients, the variable coefficient causes mathematical difficulties of the stabilization problem. By the approach of Riesz basis property, it is shown that the closed‐loop system is stable exponentially. Numerical simulations demonstrate the effect of the stable controller. This paper is devoted to the wave equation with variable coefficients generalized of that with constant coefficients for delayed observation and non‐collocated control.     

Decay of solutions for a semilinear system of elastic waves in an exterior domain with damping near infinity

We show that the solutions of a semilinear system of elastic waves in an exterior domain with a localized damping near infinity decay in an algebraic rate to zero. We impose an additional condition on the Lamé coefficients. It seems that this restriction cannot be overcome by using the two-finite-speed propagation of the elastic model, since we do not assume compact support on the initial data and because the dissipation does not have compact support. The decay rates obtained for the total energy of the linear problem and the L²$L^2$-norm of the solution improve previous results. For the semilinear problem the decay rates in this paper seem to be the first contribution, mainly in the context of initial data without compact support and localized dissipation.     

Strongly damped wave problems: Bootstrapping and regularity of solutions

The aim of the article is to present a unified approach to the existence, uniqueness and regularity of solutions to problems belonging to a class of second order in time semilinear partial differential equations in Banach spaces. Our results are applied next to a number of examples appearing in literature, which fall into the class of strongly damped semilinear wave equations. The present work essentially extends the results on the existence and regularity of solutions to such problems. Previously, these problems have been considered mostly within the Hilbert space setting and with the main part operators being selfadjoint. In this article we present a more general approach, involving sectorial operators in reflexive Banach spaces.     

Event-triggered boundary stabilization for coupled semilinear reaction–diffusion systems with spatially varying coefficients

This paper aims to address the event-triggered Robin boundary control problem for exponential stabilization of the coupled semilinear reaction–diffusion systems with spatially varying coefficients. The main used method is the backstepping, which allows us to explicitly give the boundary control formulae. More precisely, we first explore the existence and uniqueness of classical solutions for the considered problem. After this, we propose an event-triggered boundary feedback control law to exponentially stabilize the system under consideration with the Zeno phenomenon being excluded. A numerical result is finally included to illustrate the efficiency of our designed controller.     

Energy method in the partial Fourier space and application to stability problems in the half space

The energy method in the Fourier space is useful in deriving the decay estimates for problems in the whole space Rn. In this paper, we study half space problems in and develop the energy method in the partial Fourier space obtained by taking the Fourier transform with respect to the tangential variable x^′∈Rn−1. For the variable x₁∈R₊ in the normal direction, we use L² space or weighted L² space. We apply this energy method to the half space problem for damped wave equations with a nonlinear convection term and prove the asymptotic stability of planar stationary waves by showing a sharp convergence rate for t→∞. The result obtained in this paper is a refinement of the previous one in Ueda et al. (2008) [13].     

The Neumann Problem for Semilinear Elliptic Equations with Critical Sobolev Exponent

In this survey article we discuss the existence and the properties of least energy solutions of a semilinear critical Neumann problem. The main focus is on the joint effect of the shape of the graph of coefficients of the critical nonlinearities and the geometry of the boundary on the existence of solutions. Received: July 2006     

Blow-up theorem for semilinear wave equations with non-zero initial position

One of the features of solutions of semilinear wave equations can be found in blow-up results for non-compactly supported data. In spite of finite propagation speed of the linear wave, we have no global in time solution for any power nonlinearity if the spatial decay of the initial data is weak. This was first observed by Asakura (1986) [2] finding out a critical decay to ensure the global existence of the solution. But the blow-up result is available only for zero initial position having positive speed.In this paper the blow-up theorem for non-zero initial position by Uesaka (2009) [22] is extended to higher-dimensional case. And the assumption on the nonlinear term is relaxed to include an example, |u|^p−1u. Moreover the critical decay of the initial position is clarified by example.     

Global s-solvability and global s-hypoellipticity for certain perturbations of zero order of systems of constant real vector fields

The main purpose of this paper is to show, in the two-dimensional torus, a necessary and sufficient condition in order to certain perturbations of zero order of a system of constant real vector fields to be globally s-solvable. We are also interested in studying its global s-hypoellipticity. We present connections between these global concepts and a priori estimates. We also present two applications of our results for systems of operators with variable coefficients.