Quasilinear Wave Equations Research Articles

Exact boundary controllability of nodal profile for quasilinear wave equations in a planar tree‐like network of strings

Based on the theory of semi‐global piecewise C2 solutions to 1D quasilinear wave equations, the local exact boundary controllability of nodal profile for quasilinear wave equations in a planar tree‐like network of strings with general topology is obtained by a constructive method. The principles of providing nodal profiles and of choosing and transferring boundary controls are presented, respectively. Copyright © 2013 John Wiley & Sons, Ltd.

Exact boundary controllability and exact boundary observability for a coupled system of quasilinear wave equations

Based on the theory of semi-global classical solutions to quasilinear hyperbolic systems, the authors apply a unified constructive method to establish the local exact boundary (null) controllability and the local boundary (weak) observability for a coupled system of 1-D quasilinear wave equations with various types of boundary conditions.

Exponential Decay of Global Solutions for some Quasilinear Higher-Order Wave Equation

In this paper we prove the uniform stabilization of global solutions for some quasilinear higher-order wave equation with linear damping term and source term by applying a lemma due to V.Komornik.

Uniform decay rate estimates for a class of quasi-linear hyperbolic equations with non-linear damping and source terms

We consider the quasi-linear wave equation with non-linear damping and source terms where a, b, α, m, p ≥ 0. In this article, by using the multiplier methods, we give the precise uniform estimation of the decay rate, when the initial data is in the potential well. More precisely, when m = 0, the energy of solution decays exponentially and when m > 0, the energy of solution decays polynomially.

Justification of a nonlinear Schrödinger model for laser beams in photopolymers

A nonstationary model that relies on the spatial nonlinear Schrodinger (NLS) equation with the time-dependent refractive index describes laser beams in photopolymers. We consider a toy problem, when the rate of change of refractive index is proportional to the squared amplitude of the electric field and the spatial domain is a plane. After formal derivation of the NLS approximation from a two-dimensional quasilinear wave equation, we establish local well-posedness of the original and reduced models and perform rigorous justification analysis to control smallness of the approximation error for appropriately small times.

Global existence of solutions to some quasilinear wave equation in one space dimension

Under some conditions on the initial data, we show the global existence of solutions of the Cauchy problem of the quasilinear wave equation: $\partial_{t}^2 u = c(u)^2 \partial^2 _x u+\lambda c(u)c'(u)(\partial_x u)^2$ for $0 \leq \lambda\leq 2$, which has richly physical background. In [16], P. Zhang and Y. Zheng show the global existence of solutions to this equation with $\lambda =1$. It is difficult to apply their method to this global existence problem of this equation with $\lambda =0$ directly. As an application of the theorem of the global existence of solutions to $\partial_{t}^2 u = c(u)^2 \partial^2 _x u$, we construct solutions to $\partial_{t}^2 u = (u+1)^2 \partial^2 _x u$ which is going to $-1$ in finite time with $u(0,x)+1\geq \delta $ for some $\delta>0 $.

Long-time behavior of a quasilinear viscoelastic equation with past history

This paper is concerned with a class of quasilinear wave equations with memory|ut|ρutt−αΔu−Δutt+∫τtμ(t−s)Δu(s)ds−γΔut+f(u)=h,ρ>0, which was considered by several authors, with τ=0, since 2001. Existing results are mainly devoted to global existence, energy decay, existence with small data and blow-up of solutions. However uniqueness seems to be an open problem and existence of attractors was no yet considered. The objective of the present paper is to provide some results on the well-posedness and long-time behavior to this equation in a more general setting which includes past history, that is, by taking τ=−∞ in the memory term.

Numerical evidence for the validity of the NLS approximation in systems with a quasilinear quadratic nonlinearity

AbstractThe Nonlinear Schrödinger (NLS) equation is a universal amplitude equation which can be derived for the description of small spatio‐temporal modulations of an underlying carrier wave. The validity of this approximation for general dispersive wave systems possessing a quasilinear quadratic nonlinearity is an open problem that has remained unsolved for more than four decades. We report results on a systematic numerical simulation of such a quasilinear wave equation which strongly support the validity claim.

A general stability result for a quasilinear wave equation with memory

In this paper we consider a quasilinear viscoelastic problem and prove an explicit and general decay rate result, using some properties of the convex functions. Our result is established without imposing restrictive assumptions on the behavior of the relaxation function at infinity.

Remarks on global solutions for nonlinear wave equations under the standard null conditions

A combination of some weighted energy estimates is applied for the Cauchy problem of quasilinear wave equations with the standard null conditions in three spatial dimensions. Alternative proofs for global solutions are shown including the exterior domain problems.

Existence of solutions to strongly damped quasilinear wave equations

In this paper, we study the strongly damped quasilinear wave equation. By using spatial sequence techniques and energy estimate methods, we obtain the existence theorem of the solution to abstract a strongly damped wave equation and to a class of strongly damped quasilinear wave equations.

The lifespan for 3-dimensional quasilinear wave equations in exterior domains

Abstract This article focuses on long-time existence for quasilinear wave equations with small initial data in exterior domains. The nonlinearity is permitted to fully depend on the solution at the quadratic level, rather than just the first and second derivatives of the solution. The corresponding lifespan bound in the boundaryless case is due to Lindblad, and Du and Zhou first proved such long-time existence exterior to star-shaped obstacles. Here we relax the hypothesis on the geometry and only require that there is a sufficiently rapid decay of local energy for the linear homogeneous wave equation, which permits some domains that contain trapped rays. The key step is to prove useful energy estimates involving the scaling vector field for which the approach of the second author and Sogge provides guidance.

Life-span of classical solutions to one dimensional initial–boundary value problems for general quasilinear wave equations with Robin boundary conditions

This paper is devoted to studying the initial–boundary value problem for one dimensional general quasilinear wave equations utt−uxx=b(u,Du)uxx+2a0(u,Du)utx+F(u,Du) with Robin boundary conditions on an exterior domain. We obtain the sharp lower bound of the life-span of classical solutions to the initial–boundary value problem with small initial data and zero boundary data for one dimensional general quasilinear wave equations. Our result in the general case and the special case is shorter than that of the initial–boundary value problem for one dimensional general quasilinear wave equations with Dirichlet boundary conditions. The results in this paper are not the trivial generalization of that in the case of Dirichlet boundary conditions. The lower bound estimates of life span of classical solutions to initial–boundary value problems are consistent with the actual physical meaning. The physical phenomenon also explains our results.

Decay rate of quasilinear wave equation with viscosity

This paper is concerned with the decay rate of solutions for a quasilinear wave equation with viscosity. We use a so-called energy perturbation method to establish decay rate of solutions in terms of energy norm for a class of nonlinear functions. With the help of a comparison lemma of differential inequalities, we establish a relationship between decay rate of solutions and f.

Conservation laws and symmetries of quasilinear radial wave equations in multi-dimensions

Symmetries and conservation laws are studied for two classes of physically and analytically interesting radial wave equations with power nonlinearities in multi-dimensions. The results consist of two main classifications: all symmetries of point type and all conservation laws of a general energy-momentum type are explicitly determined, including those such as dilations, inversions, similarity energies, and conformal energies that exist only for special powers or dimensions. In particular, all variational cases (when a Lagrangian formulation exists) and non-variational cases (when no Lagrangian exists) for these wave equations are considered. As main results, the classification yields generalized energies and radial momenta in certain non-variational cases, which are shown to arise from a new type of Morawetz dilation identity that produces conservation laws for each of the two classes of wave equations in a different way than Noether's theorem.

The lifespan for 3-dimensional quasilinear wave equations in exterior domains

This article focuses on long-time existence for quasilinear wave equations with small initial data in exterior domains. The nonlinearity is permitted to fully depend on the solution at the quadratic level, rather than just the first and second derivatives of the solution. The corresponding lifespan bound in the boundaryless case is due to Lindblad, and Du and Zhou first proved such long-time existence exterior to star-shaped obstacles. Here we relax the hypothesis on the geometry and only require that there is a sufficiently rapid decay of local energy for the linear homogeneous wave equation, which permits some domains that contain trapped rays. The key step is to prove useful energy estimates involving the scaling vector field for which the approach of the second author and Sogge provides guidance.

On almost global existence and local well posedness for some 3-D quasi-linear wave equations

We study the Cauchy problem for a quasilinear wave equation with low-regularity data. A space-time $L^2$ estimate for the variable coefficient wave equation plays a central role for this purpose. Assuming radial symmetry, we establish the almost global existence of a strong solution for every small initial data in $H^2 \times H^1$. We also show that the initial-value problem is locally well posed.

Blow-up for a non-degenerate non-local quasilinear wave equation on IR N

We study on the global existence, decay properties and blow-up results of the solution for the following non-degenerate nonlinear wave equation where we have b ≥ 0, p ≥ 0, γ ≥ 1, δ > 0, φ(x) > 0 for all x∈ and a > 0.

Blowup and Blowup at Infinity for Quasilinear Wave Equations

We consider in this paper 3D quasilinear wave equations satisfying the null condition, for which we construct classical solutions blowing up at infinity. This is done using conformal inversion and local techniques for constructing blowup solutions. It turns out that we are dealing with non genuinely nonlinear points; hence, we are forced to consider weakly nonlinear points, the definition of which involves a new set of invariant quantities. These new invariants are of interest in themselves, and we extend their definition to first-order general systems as well.

Stability of planar diffusion wave for nonlinear evolution equation

It is known that the one-dimensional nonlinear heat equation \(u_t = f(u)_{x_1 x_1 }\), f′(u) > 0, u(±∞, t) = u−, u+ ≠ u_ has a unique self-similar solution \(\bar u\left( {\tfrac{{x_1 }} {{\sqrt {1 + t} }}} \right)\). In multi-dimensional space, \(\bar u\left( {\tfrac{{x_1 }} {{\sqrt {1 + t} }}} \right)\) is called a planar diffusion wave. In the first part of the present paper, it is shown that under some smallness conditions, such a planar diffusion wave is nonlinearly stable for the nonlinear heat equation: ut − Δf(u) = 0, x ∈ ℝn. The optimal time decay rate is obtained. In the second part of this paper, it is further shown that this planar diffusion wave is still nonlinearly stable for the quasilinear wave equation with damping: utt + ut − Δf(u) = 0, x ∈ ℝn. The time decay rate is also obtained. The proofs are given by an elementary energy method.