Semilinear Wave Equations Research Articles

On a strongly damped semilinear wave equation with time-varying source and singular dissipation

Abstract This paper deals with the global well-posedness and blow-up phenomena for a strongly damped semilinear wave equation with time-varying source and singular dissipative terms under the null Dirichlet boundary condition. On the basis of cut-off technique, multiplier method, contraction mapping principle, and the modified potential well method, we establish the local well-posedness and obtain the threshold between the existence and nonexistence of the global solution (including the critical case). Meanwhile, with the aid of modified differential inequality technique, the blow-up result of the solutions with arbitrarily positive initial energy and the lifespan of the blow-up solutions are derived.

Existence, uniqueness, and stability of Fourier series solutions of stochastic wave equations with cubic nonlinearities in 3D

Existence, uniqueness, and stability of approximate Fourier series solutions u to damped and undamped, semi-linear stochastic wave equations of the form (in Itô integral sense) utt=σ2△u+B(u,ut)+G(u,ut)∂W∂t with cubic nonlinearities B(u,ut)=a1u−a2||u||2u−κut and homogeneous boundary conditions (HBCs) on general 3D cubes D=[0,lx]×[0,ly]×[0,lz] is studied. The driving Q-regular space–time noise W with linear-growth bounded, state-dependent diffusion intensities G(u,ut) is supposed to be general in space (x,y,z)∈D, but white in time t≥0. The analysis is carried out on an appropriate, separable Hilbert space (i.e. the space of Fourier coefficients) of all solutions which are driven by the eigenfunctions of Laplace operator △ subject to HBCs on the 3D domain D. The dynamics of related expected total energy functional e(t) plays a central role in our discussion, depending on diverse parameters such as diffusivity constant σ, damping κ≥0, transport coefficient a1, length parameters lx,ly,lz, diffusion constants br of G and eigenvalues αn,m,li,j,k of related covariance operator Q. Several examples illustrate the feasibility of our approach and explain our major results.

An efficient numerical simulation of the two-dimensional semilinear wave equation

An efficient numerical simulation of the two-dimensional semilinear wave equation

On a semilinear wave equation in anti-de Sitter spacetime: The critical case

In the present paper, we prove the blow-up in finite time for local solutions of a semilinear Cauchy problem associated with a wave equation in anti-de Sitter spacetime in the critical case. According to this purpose, we combine a result for ordinary differential inequalities with an iteration argument by using an explicit integral representation formula for the solution to a linear Cauchy problem associated with the wave equation in anti-de Sitter spacetime in one space dimension.

Quenching for a semi-linear wave equation for micro-electro-mechanical systems

We consider the formation of finite-time quenching singularities for solutions of semi-linear wave equations with negative power nonlinearities, such as can model micro-electro-mechanical systems. For radial initial data we obtain, formally, the existence of a sequence of quenching self-similar solutions. Also from formal aymptotic analysis, a solution to the PDE which is radially symmetric and increases strictly monotonically with distance from the origin quenches at the origin like an explicit spatially independent solution. The latter analysis and numerical experiments suggest a detailed conjecture for the singular behaviour.

Exponential decay for semilinear wave equation with localized damping in the hyperbolic space

AbstractIn this paper, we study the exponential decay of the energy associated to an initial value problem involving the wave equation on the hyperbolic space . The space is the unit disc of endowed with the Riemannian metric g given by , where and , if and , if . Making an appropriate change, the problem can be seen as a singular problem on the boundary of the open ball endowed with the euclidean metric. The proof is based on the multiplier techniques combined with the use of Hardy's inequality, in a version due to the Brezis–Marcus, which allows us to overcome the difficulty involving the singularities.

Inverse scattering for critical semilinear wave equations

We show that the scattering operator for defocusing energy critical semilinear wave equations \square u+f(u)=0, with f C-infinity and f ~ u^5, in three space dimensions, determines f.

Exact boundary controllability of 1D semilinear wave equations through a constructive approach

The exact controllability of the semilinear wave equation $$y_{tt}-y_{xx}+ f(y)=0$$ , $$x\in (0,1)$$ assuming that f is locally Lipschitz continuous and satisfies the growth condition $$\limsup _{\vert r\vert \rightarrow \infty } \vert f(r)\vert /(\vert r\vert \ln ^{p}\vert r\vert )\leqslant \beta $$ for some $$\beta $$ small enough and $$p=2$$ has been obtained by Zuazua (Ann Inst H Poincaré Anal Non Linéaire 10(1):109–129, 1993). The proof based on a non-constructive fixed point arguments makes use of precise estimates of the observability constant for a linearized wave equation. Under the above asymptotic assumption with $$p=3/2$$ , by introducing a different fixed point application, we present a simpler proof of the exact boundary controllability which is not based on the cost of observability of the wave equation with respect to potentials. Then, assuming that f is locally Lipschitz continuous and satisfies the growth condition $$\limsup _{\vert r\vert \rightarrow \infty } \vert f^\prime (r)\vert /\ln ^{3/2}\vert r\vert \leqslant \beta $$ for some $$\beta $$ small enough, we show that the above fixed point application is contracting yielding a constructive method to approximate the controls for the semilinear equation. Numerical experiments illustrate the results. The results can be extended to the multi-dimensional case and for nonlinearities involving the gradient of the solution.

Global solutions and blow-up for a class of strongly damped wave equations systems

The initial-boundary value problem for semilinear wave equation systems with a strong dissipative term in bounded domain is studied. The existence of global solutions for this problem is proved by using potential well method, and the exponential decay of global solutions is given through introducing an appropriate Lyapunov function. Meanwhile, blow-up of solutions in the unstable set is also obtained.

Semilinear wave equations of derivative type with spatial weights in one space dimension

This paper is devoted to the initial value problems for semilinear wave equations of derivative type with spatial weights in one space dimension. The lifespan estimates of classical solutions are quite different from those for nonlinearity of unknown function itself as the global-in-time existence can be established by spatial decay.

On a system of weakly null semilinear wave equations

On a system of weakly null semilinear wave equations

A reaction–diffusion approximation of a semilinear wave equation with damping

A reaction–diffusion approximation of a semilinear wave equation with damping

Existence of Global Attractors for a Semilinear Wave Equation with Nonlinear Boundary Dissipation and Nonlinear Interior and Boundary Sources with Critical Exponents

Existence of Global Attractors for a Semilinear Wave Equation with Nonlinear Boundary Dissipation and Nonlinear Interior and Boundary Sources with Critical Exponents

The Dirichlet-to-Neumann map for a semilinear wave equation on Lorentzian manifolds

We consider the semilinear wave equation on a Lorentzian manifold (M, g) with timelike boundary. We show that from the knowledge of the Dirichlet-to-Neumann map one can recover the metric g and the coefficient a up to natural obstructions. Our proof rests on the analysis of the interaction of distorted plane waves together with a scattering control argument, as well as Gaussian beam solutions.

Uniqueness, reconstruction and stability for an inverse problem of a semi-linear wave equation

We consider the recovery of a potential associated with a semi-linear wave equation on Rn+1, n≥1. We show that an unknown potential a(x,t) of the wave equation □u+aum=0 can be recovered in a Hölder stable way from the map u|∂Ω×[0,T]↦〈ψ,∂νu|∂Ω×[0,T]〉L2(∂Ω×[0,T]). This data is equivalent to the inner product of the Dirichlet-to-Neumann map with a measurement function ψ. We also prove similar stability result for the recovery of a when there is noise added to the boundary data. The method we use is constructive and it is based on the higher order linearization. As a consequence, we also get a uniqueness result. We also give a detailed presentation of the forward problem for the equation □u+aum=0.

Global existence, blow-up in finite time and vacuum isolating phenomena for a system of semilinear wave equations associated with the helical flows of Maxwell fluid

In this paper, we investigate the initial boundary value problem for the system of semilinear wave equations associated with the helical flows of Maxwell fluid. We introduce a family of potential wells and discuss the invariant sets and vacuum isolating behavior of solutions. Using the potential well argument, we show the global existence, finite time blow-up, the asymptotic behavior of solutions, estimate the lifespan as well as blow-up rate of the weak solution. In particular, we establish a sharp criterion for global existence and blow-up of solutions when E0⩽d. Finally, when the initial energy is supercritical, we give some explicit criterion for blow-up in finite.

Stabilization of the critical nonlinear Klein-Gordon equation with variable coefficients on R^3

We prove the exponential stability of the defocusing critical semilinear wave equation with variable coefficients and locally distributed damping on \(\mathbb{R}^3\). The construction of the variable coefficients is almost equivalent to the geometric control condition. We develop the traditional Morawetz estimates and the compactness-uniqueness arguments for the semilinear wave equation to prove the unique continuation result. The observability inequality and the exponential stability are obtained subsequently.

Blow-up of positive-initial-energy solutions for nonlinearly damped semilinear wave equations

Blow-up of positive-initial-energy solutions for nonlinearly damped semilinear wave equations

Blow-up of solution to semilinear wave equations with strong damping and scattering damping

This paper is devoted to investigating the initial boundary value problem for a semilinear wave equation with strong damping and scattering damping on an exterior domain. By introducing suitable multipliers and applying the test-function technique together with an iteration method, we derive the blow-up dynamics and an upper-bound lifespan estimate of the solution to the problem with power-type nonlinearity |u|^{p}, derivative-type nonlinearity |u_{t}|^{p}, and combined type nonlinearities |u_{t}|^{p}+|u|^{q} in the scattering case, respectively. The novelty of the present paper is that we establish the upper-bound lifespan estimate of the solution to the problem with strong damping and scattering damping, which are associated with the well-known Strauss exponent and Glassey exponent.

A large data theory for nonlinear wave on the Schwarzschild background

We study both of the scattering and Cauchy problems for the semilinear wave equation with a null quadratic form on the Schwarzschild background. Prescribing the scattering data that are given by the short pulse data on the future null infinity and are trivial on the future event horizon, we construct a class of global solutions backwards up to any finite time and show that the wave travels in such a way that almost all of the (large) energy is focusing in an outgoing null strip, while little radiates out of this strip. In reverse, considering a class of Cauchy data with large energy norms, there exists a unique and global solution in the future development, and the radiation field along the future null infinity exists. Furthermore, most of the wave packet is confined in an incoming null strip and reflected to the future event horizon, whereas little is transmitted to the future null infinity.