Related papers: Asymptotic behaviour of non-radiative solution to …
In this work we consider the energy subcritical 3D wave equation $\partial_t^2 u - \Delta u = \pm |u|^{p-1} u$ and discuss its (weakly) non-radiative solutions, i.e. the solutions defined in an exterior region $\{(x,t): |x|>|t|+R\}$ with…
Radiation field and channel of energy method have become important tools in the study of nonlinear wave equations in recent years. In this work we give basic theory of radiation fields of free waves in the energy sub-critical case. We also…
In this work we classify all radial non-radiative solutions to the 5D nonlinear wave equations with a wide range of energy critical nonlinearity. We show that such a solution always comes with two characteristic numbers. These…
Non-radiative solutions of energy critical wave equations are such that their energy in an exterior region $|x|>R+|t|$ vanishes asymptotically in both time directions. This notion, introduced by Duyckaerts, Kenig and Merle (J. Eur. Math.…
We discuss a notion of weak solution for a semilinear wave equation that models the interaction of an elastic body with a rigid substrate through an adhesive layer, relying on results in [2]. Our analysis embraces the vector-valued case in…
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…
In this work we consider a wide range of energy critical wave equation in 3-dimensional space with radial data. We are interested in exterior scattering phenomenon, in which the asymptotic behaviour of a solutions $u$ to the non-linear wave…
In this paper, we study the asymptotic behavior of radial solutions for several weighted elliptic equations with power type or exponential type nonlinearities on an annulus.
We present a scaling technique which transforms the evolution problem for a nonlinear wave equation with small initial data to a linear wave equation with a distributional source. The exact solution of the latter uniformly approximates the…
Non radiative solutions of the energy critical non linear wave equation are global solutions $u$ that furthermore have vanishing asymptotic energy outside the lightcone at both $t \to \pm \infty$:\[ \lim_{t \to \pm \infty} \| \nabla_{t,x}…
We present a new method for constructing solutions to nonlinear evolutionary equations describing the propagation and interaction of nonlinear waves.
We consider a class of scalar quasilinear wave equations in three spatial dimensions satisfying the weak null condition. For solutions arising from small, localized, smooth data, we give an asymptotic formula describing the global…
We give a proof for the existence of a weak solution on the initial-value problem of a non-linear damped propagation
Spiral wave solutions are found in linear and weakly nonlinear irrotational water wave equations. These unsteady spiral waves evolve from suitable initial conditions; they are not induced by external forcing. In the linear case, a long-time…
We establish spectral, linear, and nonlinear stability of the vanishing and slow-moving travelling waves that arise as time asymptotic solutions to the Fisher-Stefan equation. Nonlinear stability is in terms of the limiting equations that…
In this paper, we prove the first asymptotic completeness result for a scalar quasilinear wave equation satisfying the weak null condition. The main tool we use in the study of this equation is the geometric reduced system introduced in…
Consider the energy-critical focusing wave equation in space dimension $N\geq 3$. The equation has a nonzero radial stationary solution $W$, which is unique up to scaling and sign change. It is conjectured (soliton resolution) that any…
We establish precise asymptotic expansions for solutions to semilinear wave equations with power-type nonlinearities on asymptotically flat spacetimes. Our analysis focuses on two key cases: cubic nonlinearities and higher-order power…
We give asymptotics for Einstein vacuum equations in wave coordinates with small asymptotically flat data. We show that the behavior is wave like at null infinity and homogeneous towards time like infinity. We use the asymptotics to show…
We consider a nonlinear parabolic equation with an exponential nonlinearity which is critical with respect to the growth of the nonlinearity and the regularity of the initial data. After showing the equivalence of the notions of weak and…