Related papers: Global existence for 2-D wave maps equation in ext…
We show a global existence result for a doubly nonlinear porous medium type equation of the form $$u_t = \Delta_p u^m +\, u^q$$ on a complete and non-compact Riemannian manifold $M$ of infinite volume. Here, for $1<p<N$, we assume…
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 $(\partial_u^2 Q)(0,0,0)=0$.…
For the $2$-D semilinear wave equation with scale-invariant damping $\square u+\frac{\mu}{t}\partial_tu=|u|^p$, where $t\geq 1$, $\mu>0$ and $p>1$, it is conjectured that the global small data weak solution $u$ exists when $p>p_{s}(2+\mu)…
In this paper, we show global existence, in spatial dimensions greater than or equal to four, for semilinear wave equations with quadratic nonlinearities exterior to a nontrapping obstacle. This extends the previous work of Shibata-Tsutsumi…
The paper is devoted to investigating long time behavior of smooth small data solutions to 3-D quasilinear wave equations outside of compact convex obstacles with Neumann boundary conditions.
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…
Assuming initial data have small weighted $H^4\times H^3$ norm, we prove global existence of solutions to the Cauchy problem for systems of quasi-linear wave equations in three space dimensions satisfying the null condition of Klainerman.…
We study the global existence of solutions to semilinear wave equations with power-type nonlinearity and general lower order terms on $n$ dimensional nontrapping asymptotically Euclidean manifolds, when $n=3, 4$. In addition, we prove…
For any subcritical index of regularity $s>3/2$, we prove the almost global well posedness for the 2-dimensional semilinear wave equation with the cubic nonlinearity in the derivatives, when the initial data are small in the Sobolev space…
In this paper we consider the following Cauchy problem for the semi-linear wave equation with scale-invariant dissipation and mass and power non-linearity: \begin{align}\label{CP abstract} \begin{cases} u_{tt}-\Delta u+\dfrac{\mu_1}{1+t}…
In this paper we prove global existence and global behavior of solutions to quasilinear wave-Klein-Gordon systems in $\mathbb{R}^{1+2}$ with quadratic nonlinearities satisfying the null condition. We consider small, regular and compactly…
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…
This paper is devoted to the proof of a global existence result for the water waves equation with smooth, small, and decaying at infinity Cauchy data. We obtain moreover an asymptotic description in physical coordinates of the solution,…
This paper proves global existence and sharp pointwise decay for solutions to nonlinear wave equations satisfying the semilinear null condition, on a class of three-dimensional, asymptotically flat, and notably, non-stationary spacetimes.…
We prove almost global existence for multiple speed quasilinear wave equations with quadratic nonlinearities in three spatial dimensions. We prove new results both for Minkowski space and also for nonlinear Dirichlet-wave equations outside…
The aim of this article is to present an elementary proof of a global existence result for nonlinear wave equations satifying the null condition in an exterior domain. The novelty of our proof is to avoid completely the scaling operator…
We consider the global existence and blow up of solutions of the Cauchy problem of the quasilinear wave equation: $\partial_{t}^2 u = \partial_x(c(u)^2 \partial_x u)$, which has richly physical backgrounds. Under the assumption that…
We discuss how the higher-order term $|u|^q$ $(q>1+2/(n-1))$ has nontrivial effects in the lifespan of small solutions to the Cauchy problem for the system of nonlinear wave equations $$ \partial_t^2 u-\Delta u=|v|^p, \qquad \partial_t^2…
Lifespan estimates for semilinear damped wave equations of the form $\partial_t^2u-\Delta u+\partial_tu=|u|^p$ in a two dimensional exterior domain endowed with the Dirichlet boundary condition are dealt with. For the critical case of the…
Our interest itself of this paper is strongly inspired from an open problem in the paper [1] published by D'Abbicco. In this article, we would like to study the Cauchy problem for a weakly coupled system of semi-linear structurally damped…