Class Of Semilinear Wave Equations Research Articles

Periodic solutions with irrational frequency for a class of semilinear wave equations with variable coefficients

This paper is devoted to the study of the existence of periodic solutions for a class of semilinear wave equations with variable coefficients. The forced vibrations of a nonhomogeneous string and propagation of seismic waves in nonisotropic media is governed by this mathematical model. When the frequency is a sufficiently large irrational number with bounded partial quotients, the existence of weak solutions is established. Then, under some suitable conditions, we improve the regularity of weak solutions. Our results can also be applied to the corresponding constant coefficients wave equation.

Lifespan Estimates for a Class of Semilinear Wave Equations with Time- and Space-Dependent Coefficients on the Power Nonlinearity

Lifespan Estimates for a Class of Semilinear Wave Equations with Time- and Space-Dependent Coefficients on the Power Nonlinearity

Energy decay for small solutions to semilinear wave equations with weakly dissipative structure

This article gives an energy decay result for small data solutions to a class of semilinear wave equations in two space dimensions possessing weakly dissipative structure relevant to the Agemi condition.

Blowup behavior for strongly perturbed wave equations

We study the blowup behavior of a class of strongly perturbed wave equations with a focusing supercritical power nonlinearity in three spatial dimensions. We show that the ODE blowup profile of the unperturbed equation still describes the asymptotics of stable blowup. As a consequence, stable ODE-type blowup is seen to be a universal phenomenon that exists in a large class of semilinear wave equations.

Solutions with a uniform time of existence of a class of characteristic semi-linear wave equations near

We prove existence and uniqueness of solution of a class of semi-linear wave equations with initial data prescribed on the light-cone with vertex the origin of a Minkowski space-time. The nonlinear term is assumed to satisfy a nullity condition which guarantees that the neighborhood of the initial cone on which we obtain our solution does not shrink to zero as one approaches infinity. This result is applied to wave maps on Minkowski space-time with

General Stability and Exponential Growth for a Class of Semi-linear Wave Equations with Logarithmic Source and Memory Terms

In this work we investigate asymptotic stability and instability at infinity of solutions to a logarithmic wave equation $$\begin{aligned} u_{tt}-\Delta u + u + (g\,*\, \Delta u)(t)+ h(u_{t})u_{t}+|u|^{2}u=u\log |u|^{k}, \end{aligned}$$ in an open bounded domain $$\Omega \subseteq \mathbb {R}^3$$ whith $$h(s)=k_{0}+k_{1}|s|^{m-1}.$$ We prove a general stability of solutions which improves and extends some previous studies such as the one by Hu et al. (Appl Math Optim, https://doi.org/10.1007/s00245-017-9423-3 ) in the case $$g=0$$ and in presence of linear frictional damping $$u_{t}$$ when the cubic term $$|u|^2u$$ is replaced with u. In the case $$k_{1}=0,$$ we also prove that the solutions will grow up as an exponential function. Our result shows that the memory kernel g dose not need to satisfy some restrictive conditions to cause the unboundedness of solutions.

Optimal decay rates for semilinear wave equations with memory and Neumann boundary conditions

We consider a class of semilinear wave equations with memory and Neumann boundary conditions, being subject to frictional dissipation. As can be seen, their solutions have different properties from those in the case of Dirichlet boundary conditions (or Dirichlet–Neumann boundary conditions). We prove for some nonlinear terms with growth exponent that all solutions decay uniformly and at least at the polynomial rate , when the memory kernel decays exponentially or polynomially (with large enough degree r); some other decay rates, depending on both q and r, are also derived, when r is not large enough. Moreover, we show a large class of solutions decaying exactly at the rate , in the case of the memory kernels decaying exponentially.

Global existence and finite time blow up for a class of semilinear wave equations on [formula omitted

In this paper, we consider the initial value problem for a class of semilinear wave equations with damping and source terms. Under certain conditions we establish the global existence of solutions, and the asymptotic behavior of the solutions by introducing an appropriate Lyapunov functions. Moreover, we examine the blow-up in finite time when the initial data is sufficiently large.

Global stability of solutions to nonlinear wave equations

We consider the problem of global stability of solutions to a class of semilinear wave equations with null condition in Minkowski space. We give sufficient conditions on the given solution, which guarantees stability. Our stability result can be reduced to a small data global existence result for a class of semilinear wave equations with linear terms $$B^{\mu \nu }\partial _\mu \Phi (t, x)\partial _\nu \phi $$ , $$L^\mu (t,x)\partial _\mu \phi $$ and quadratic terms $$h^{\mu \nu }(t, x)\partial _\mu \phi \partial _\nu \phi $$ where the functions $$\Phi (t, x)$$ , $$L^\mu (t, x)$$ , $$h^{\mu \nu }(t, x)$$ decay rather weakly and the constants $$B^{\mu \nu }$$ satisfy the null condition. We show the small data global existence result by using the new approach developed by Dafermos–Rodnianski. In particular, we prove the global stability result under weaker assumptions than those imposed by Alinhac (Indiana Univ Math J 58(6):2543–2574, 2009).

Global Solutions of Nonlinear Wave Equations in Time Dependent Inhomogeneous Media

We consider the problem of small data global existence for a class of semilinear wave equations with null condition on a Lorentzian background $(\mathbb{R}^{3+1}, g)$ with a \textbf{time dependent metric $g$} coinciding with Minkowski metric outside the cylinder $\{(t, x)| |x|\leq R\}$. We show that the small data global existence result can be reduced to two integrated local energy estimates and demonstrate these estimates in the particular case when $g$ is merely $C^1$ close to the Minkowski metric. One of the novel aspects of this work is that it applies to equations on backgrounds which do not settle to any particular stationary metric.

Nonlinear instability of scalar fields on extremal black holes

We prove a new type of finite time blow-up for a class of semilinear wave equations on extremal black holes. The initial data can be taken to be arbitrarily close to the trivial data. The first singularity occurs along the (degenerate) future event horizon. No analogue of this instability occurs for subextremal black holes or the Minkowski spacetime.

The energy decay and asymptotics for a class of semilinear wave equations in two space dimensions

We consider semilinear wave equations with small initial data in two space dimensions. For a class of wave equations with cubic nonlinearity, we show the global existence of small amplitude solutions and give an asymptotic description of the solution as t → ∞ uniformly in $${x \in \mathbb{R}^2}$$ . In particular, our result implies the decay of the energy when the nonlinearity is dissipative.

The influence of oscillations on global existence for a class of semi-linear wave equations

The goal of this paper is to study the global existence of small data solutions to the Cauchy problem for the nonlinear wave equation In particular we are interested in statements for the 1D case. We will explain how the interplay between the increasing and oscillating behavior of the coefficient will influence global existence of small data solutions. Copyright © 2011 John Wiley & Sons, Ltd.

On the solvability of one boundary value problem for some semilinear wave equations with source terms

In a conic domain of time type for one class of semilinear wave equations with source terms we consider a Sobolev problem representing a multidimensional version of the Darboux second problem. The questions on global and local solvability, uniqueness and absence of solutions of this problem are investigated.

Interaction of kinks for semilinear wave equations with a small parameter

We consider a class of semilinear wave equations with a small parameter and nonlinearities such that the equations have exact kink-type solutions. The main result consists in obtaining sufficient conditions for the nonlinearities under which the interaction of kinks preserves the sine-Gordon scenario. This means that the interaction occurs without changing the waves shape and with shifts of trajectories.

Asymptotics of Kink--Kink Interaction for Sine-Gordon Type Equations

We consider a class of semilinear wave equations with a small parameter e. The nonlinearity of F(u) is assumed to be such that the corresponding equation has an exact self-similar solution of kink type. For F(u), we obtain sufficient conditions for two kinks to interact (in the sense of the leading term of the asymptotics with respect to e) in the same way as the kinks of the sine-Gordon equation.

Almost periodic solution of a class of semilinear wave equations with boundary dissipation

Almost periodic solution of a class of semilinear wave equations with boundary dissipation

Exponential stability of classical solutions to a class of semilinear wave equations with boundary dissipation

Exponential stability of classical solutions to a class of semilinear wave equations with boundary dissipation

Long-time existence for small amplitude semilinear wave equations

We prove that a certain class of semilinear wave equations has global solutions if the initial data is small. These existence results rely on mixed-norm angular-radial space-time estimates. More specifically, we consider power nonlinearities, □ u = F p ( u ), where F p ( u ) ~ | u | p and □ = ∂ 2 t - Δ denotes the D'Alembertian on [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]. In 1979, John investigated this equation for small initial data in dimension n = 3. He proved that if p > 1 + √2 then there is a global solution whereas for p < 1 + √2, the solution may blow up. Shortly afterwards Strauss conjectured that a similar result should hold in n dimensions, and the critical power p c above which global solutions exist should be the positive root of ( n -1) p 2 c -( n +1) p c - 2 = 0. Sideris proved that for all n the solutions can blow up if p < p c . Recently Zhou proved that for n = 4 one has global solutions if p > p c = 2. Here we prove that for n ≤ 8 one has global solutions if p > p c . Furthermore, we proved that for all n , one has global solutions if p > p c and the initial data are spherically symmetric. In the radial case we use L p estimates, derived from the explicit form of the fundamental solution using inequalities from classical analysis. In the nonradial case we write the fundamental solution in polar coordinates and use Fourier integral and special function estimates to handle the angular part.

On multiple solutions for a class of semilinear wave equations

On multiple solutions for a class of semilinear wave equations