Solutions Of Quasilinear Wave Equation Research Articles

Quasilinear wave equations on Schwarzschild–de Sitter

We give an elementary new argument for global existence and exponential decay of solutions of quasilinear wave equations on Schwarzschild–de Sitter black hole backgrounds, for appropriately small initial data. The core of the argument is entirely local, based on time translation invariant energy estimates in spacetime slabs of fixed time length. Global existence then follows simply by iterating this local result in consecutive spacetime slabs. We infer that an appropriate future energy flux decays exponentially with respect to the energy flux of the initial data.

General decay of solutions of quasilinear wave equation with time‐varying delay in the boundary feedback and acoustic boundary conditions

In this paper, we are concerned with the general decay result of the quasi‐linear wave equation with a time‐varying delay in the boundary feedback and acoustic boundary conditions. Copyright © 2017 John Wiley & Sons, Ltd.

Analyticity of solutions for quasilinear wave equations and other quasilinear systems

We prove the persistence of analyticity for classical solutions of the Cauchy problem for quasilinear wave equations with analytic data. Our results show that the analyticity of solutions, stated by the Cauchy–Kowalewski and Ovsiannikov–Nirenberg theorems, lasts until a classical solution exists. Moreover, they show that if the equation and the Cauchy data are analytic only in a part of the space variables, then a classical solution is also analytic in these variables. The approach applies to other quasilinear equations and implies the persistence of the space analyticity (and the partial space analyticity) of their classical solutions.

Solutions of semilinear wave equation via stochastic cascades

We introduce a probabilistic representation for solutions of quasilinear wave equation with analytic nonlinearities. We use stochastic cascades to prove existence and uniqueness of the solution.

Global Solutions of Quasilinear Wave Equations

We show global small data existence for a class of quasilinear wave equations related to Einstein's equations in harmonic coordinates. These equations do not satisfy the classical null condition and the asymptotic behavior of solutions is not free but the light cones bend at infinity.

Global low regularity solutions of quasi-linear wave equations

In this paper we prove the global existence and uniqueness of the low regularity solutions to the Cauchy problem of quasi-linear wave equations with radial symmetric initial data in three space dimensions. The results are based on the end-point Strichartz estimate together with the characteristic method.

A note on “On the existence of solutions of quasilinear wave equations with viscosity”

In the note, we study the existence and uniqueness of the global generalized solution for the following initial boundary value problem: (1) u t t ( t , x ) − div { σ ( | ∇ u ( t , x ) | 2 ) ∇ u ( t , x ) } − Δ u ( t , x ) − Δ u t ( t , x ) + δ | u t ( t , x ) | p − 1 u t ( t , x ) = μ | u ( t , x ) | q − 1 u ( t , x ) , x ∈ Ω , t ∈ ( 0 , T ) , (2) u ( t , x ) | ∂ Ω = 0 , t ∈ ( 0 , T ) , (3) u ( 0 , x ) = u 0 ( x ) , u t ( 0 , x ) = u 1 ( x ) , x ∈ Ω , where Ω is a bounded domain in R N with smooth boundary ∂ Ω , p ≥ 1 , q > 1 , δ > 0 , μ > 0 , Δ denotes the Laplacian operator in R N , ∇ denotes the gradient operator and σ ( s ) is a given nonlinear function.

Modulating pulse solutions for quasilinear wave equations

This paper presents an existence proof for symmetric modulating pulse solutions of a quasilinear wave equation. Modulating pulse solutions consist of a pulse-like envelope advancing in the laboratory frame and modulating an underlying wave train; they are also referred to as ‘moving breathers’ since they are time periodic in a moving frame of reference. The problem is formulated as an infinite-dimensional dynamical system with two stable, two unstable and infinitely many neutral directions. Using a partial normal form and a generalisation of local invariant-manifold theory to the quasilinear setting we prove the existence of modulating pulses on arbitrarily large, but finite domains in space and time.

Remarks on energy inequalities for wave and Maxwell equations on a curved background

In our paper [1] establishing lower bounds for the lifespan of classical solutions of quasilinear wave equations satisfying the null condition, we introduce the method of “ghost weight” to obtain a better control of some special derivatives in the energy inequalities. Such an improvement has turned out to be useful in applications to nonlinear problems, see for instance [2]. Hence we thought of making the method more explicit and more systematic, which is the aim of the present paper. The Theorems we propose in this paper are mere illustrations of what can be done, and many variations are possible. The main difference between our inequalities and the standard ones is that they display the better behavior of the special derivatives Ti = ∂i + ωi∂t of a given solution φ. These derivatives span the tangent planes to the cones r− t = C at any given point, and are related to the Z-fields used by Klainerman in his approach of decaying solutions to wave equations. Hence all tangential components of ∇φ = (∂tφ, ∂xφ) behave in fact better than what is given by the standard energy inequality: we say that we are in a “1D-situation”. This fact has an important consequence for the amplification factor of the inequality. By this, we mean the term

Global existence of solutions for quasi-linear wave equations with viscous damping

In this paper, the global existence of solutions to the initial boundary value problem for a class of quasi-linear wave equations with viscous damping and source terms is studied by using a combination of Galerkin approximations, compactness, and monotonicity methods.

On solutions of quasilinear wave equations with nonlinear damping terms

In this paper we consider the existence and asymptotic behavior of solutions of the following problem: $$\begin{gathered} utt(t,x) - (\alpha + \beta {\kern 1pt} \parallel \,\nabla u(t,x)\,\parallel _2^2 + \beta \,\parallel \,\nabla u(t,x)\,\parallel _2^2 )\Delta u(t,x) + \delta \,|\,u_t (t,x)^{p - 1} u_t (t,x) \hfill \;\;\; = \mu \,|\,u(t,x){\kern 1pt} \,|^{q - 1} u(t,x),\;\;\;x \in \Omega ,\;\;\;t \geqslant 0, \hfill \end{gathered}$$ $$\begin{gathered} utt(t,x) - (\alpha + \beta {\kern 1pt} \parallel \,\nabla u(t,x)\,\parallel _2^2 + \beta \,\parallel \,\nabla {\kern 1pt} \nu (t,x)\,\parallel _2^2 )\Delta \nu (t,x) + \delta \,|\,\nu _t (t,x)^{p - 1} \nu _t (t,x) \hfill \;\;\; = \mu \,|\,\nu (t,x){\kern 1pt} \,|^{q - 1} \nu (t,x),\;\;\;x \in \Omega ,\;\;\;t \geqslant 0, \hfill \end{gathered}$$ $$u(0,x) = u_0 (x),\;\;u_t (0,x) = u_1 (x),\;\;x \in \Omega ,$$ $$v(0,x) = v_0 (x),\;\;v_t (0,x) = v_1 (x),\;\;x \in \Omega ,$$ $$u{\kern 1pt} |_{\;\partial \Omega } = v{\kern 1pt} |_{\;\partial \Omega } = 0$$ where q>1, q⩾1, δ>0, α>0, β⩾0, \(\mu \in {\mathbb{R}}\) is the Laplacian in \({\mathbb{R}}^N\).

Rank 2 singular solutions for quasilinear wave equations

For quasilinear wave equations with analytic coefficients, we construct singular solutions u such that u;u0 remain continuous, while u00 is singular like (T − t)−1 and has rank 2. This work extends our previous work on geometric blowup to mere multidimensional solutions.

Formation of singularities for wave equations including the nonlinear vibrating string

AbstractUnder very general assumptions, the authors prove that smooth solutions of quasilinear wave equations with small‐amplitude periodic initial data always develop singularities in the second derivatives in finite time. One consequence of these results is the fact that all solutions of the classical nonlinear vibrating string equation satisfying either Dirichlet or Neumann boundary conditions and with sufficiently small nontriviai initial data necessarily develop singularities. In particular, there are no nontrivial smooth small‐amplitude time‐periodic solutions.

On the breakdown phenomena of solutions of quasilinear wave equations.

On the breakdown phenomena of solutions of quasilinear wave equations.

Existence and nonexistence in the large of solutions of quasilinear wave equations

Existence and nonexistence in the large of solutions of quasilinear wave equations