Related papers: On one-dimension quasilinear wave equations with n…
We give an alternative proof of the global existence result originally due to Hidano and Yokoyama for the Cauchy problem for a system of quasi-linear wave equations in three space dimensions satisfying the weak null condition. The feature…
Systems of wave equations may fail to be globally well posed, even for small initial data. Attempts to classify systems into well and ill-posed categories work by identifying structural properties of the equations that can work as…
For the short pulse initial data with a first order outgoing constraint condition and optimal orders of smallness, we establish the global existence of smooth solutions to 2D quasilinear wave equations with higher order null conditions.…
In this paper we prove global existence for certain multispeed Dirichlet-wave equations with quadratic nonlinearities outside of obstacles. We assume the natural null condition for systems of quasilinear wave equations with multiple speeds.…
We establish global existence in 3+1 dimensions of small-amplitude solutions of quasilinear Dirichlet-wave equations satisfying the null condition outside of star-shapped obstacles.
We study two-dimensional semilinear strongly damped wave equation with mixed nonlinearity $|u|^p+|u_t|^q$ in an exterior domain, where $p,q>1$. Assuming the smallness of initial data in exponentially weighted spaces and some conditions on…
In this paper, we investigate the lifespan estimates of classical solutions of the initial value problems for semilinear wave equations of derivative type with characteristic weights in one space dimension. Such equations provide us basic…
In this paper, we establish a relation between two seemingly unrelated concepts for solving first-order hyperbolic quasilinear systems of partial differential equations in many dimensions. These concepts are based on a variant of the…
We consider a system of quasilinear wave equations on the product space $\mathbb{R}^{1+3}\times \mathbb{S}^1$, which we want to see as a toy model for Einstein equations with additional compact dimensions. We show global existence for small…
We prove the existence of the modified wave operators for a scalar quasilinear wave equation satisfying the weak null condition. This is accomplished in three steps. First, we derive a new reduced asymptotic system for the quasilinear wave…
The classical system of shallow-water (Saint--Venant) equations describes long surface waves in an inviscid incompressible fluid of a variable depth. Although shock waves are expected in this quasilinear hyperbolic system for a wide class…
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…
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…
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…
In this paper, we overview the recent progresses on the lifespan estimates of classical solutions of the initial value problems for nonlinear wave equations in one space dimension. There are mainly two directions of the developments on the…
This article concerns the time growth of Sobolev norms of classical solutions to the 3D quasi-linear wave equations with the null condition.
This article focuses on almost global existence for quasilinear wave equations with small initial data in 4-dimensional exterior domains. The nonlinearity is allowed to depend on the solution at the quadratic level as well as its first and…
We show global existence of small solutions to the Cauchy problem for a system of quasi-linear wave equations in three space dimensions. The feature of the system lies in that it satisfies the weak null condition, though we permit the…
Inspired by the work of Wang and Yu [21] on wave maps, we show that for all positive numbers T_{0} > 0 and E_{0} > 0, a large kind of semi-linear wave equation on R \times R^{3} has a solution whose life-span is [0; T_{0}], and the energy…
H\"ormander proved global existence of solutions for sufficiently small initial data for scalar wave equations in $(1+4)-$dimensions of the form $\Box u = Q(u, u', u'')$ where $Q$ vanishes to second order and $(\partial_u^2 Q)(0,0,0)=0$.…