Related papers: Generically sharp decay and blowing up at infinity…
We give a short proof of the existence of a small piece of null infinity for $(3+1)$-dimensional spacetimes evolving from asymptotically flat initial data as solutions of the Einstein vacuum equations. We introduce a modification of the…
In this paper we prove the blow-up theorem in the critical case for weakly coupled systems of semilinear wave equations in high dimensions. The upper bound of the lifespan of the solution is precisely clarified.
We study the large time behavior of solutions to the semilinear wave equation with space-dependent damping and absorbing nonlinearity in the whole space or exterior domains. Our result shows how the amplitude of the damping coefficient, the…
One method for the numerical treatment of future null-infinity is to decouple coordinates from the tensor basis and choose each in a careful manner. This dual-frame approach is hampered by logarithmically divergent terms that appear in a…
We analyze a simple example of wave equation with a time-dependent damping term, whose coefficient decays at infinity at the scale-invariant rate and includes an oscillatory component that is integrable but not absolutely integrable. We…
In this paper, a class of variable-coefficient wave equations equipped with time-dependent damping and the nonlinear source is considered. We show that the total energy of the system decays to zero with an explicit and precise decay rate…
We study the Cauchy problem of the damped wave equation \begin{align*} \partial_{t}^2 u - \Delta u + \partial_t u = 0 \end{align*} and give sharp $L^p$-$L^q$ estimates of the solution for $1\le q \le p < \infty\ (p\neq 1)$ with derivative…
We consider the total energy decay together with L^2-bound of the solution itself of the Cauchy problem for wave equations with a localized damping and a short-range potential. We treat it in the one dimensional Euclidean space R. We adopt…
In this paper, we discuss the global existence of weak solutions to the semilinear damped wave equation \begin{equation*} \begin{cases} \partial_t^2u-\Delta u + \partial_tu = f(u) & \text{in}\ \Omega\times (0,T), \\ u=0 & \text{on}\…
We prove weighted-$L^\infty$ and pointwise space-time decay estimates for weak solutions of a class of wave equations with time-independent potentials and subject to initial data, both of low regularity, satisfying given decay bounds at…
In this work, we study the finite time blow-up phenomenon of three types of semilinear wave systems with multiple speeds, posed on asymptotically Euclidean manifolds. We establish the upper bound estimates for the lifespan of solutions when…
This work is concerned with the generation of decay estimates in the velocity variable for solutions of the space-inhomogeneous Boltzmann equation without cutoff on a bounded spatial domain for hard and moderately soft potentials. We work…
We study systems of nonlinear ordinary differential equations where the dominant term, with respect to large spatial variables, causes blow-ups and is positively homogeneous of a degree $1+\alpha$ for some $\alpha>0$. We prove that the…
We provide a uniform decay estimate of Morawetz type for the local energy of general solutions to the inhomogeneous wave equation on a Schwarzchild background. This estimate is both uniform in space and time, so in particular it implies a…
In this paper, we investigate the global behaviors of solutions to defocusing semilinear wave equations in $\mathbb{R}^{1+d}$ with $d\geq 3$. We prove that in the energy space the solution verifies the integrated local energy decay…
We prove by using an iteration argument some blow-up results for a semilinear damped wave equation in generalized Einstein-de Sitter spacetime with a time-dependent coefficient for the damping term and power nonlinearity. Then, we…
In this paper, we study a semilinear weakly coupled system of wave equations with power nonlinearities. More precisely, we couple (through the nonlinear terms) a wave equation and a damped wave equation with a time-dependent coefficient for…
We consider the problem of energy decay rates for nonlinearly damped abstract infinite dimensional systems. We prove sharp, simple and quasi-optimal energy decay rates through an indirect method, namely a weak observability estimate for the…
A degenerate fourth-order parabolic equation modeling condensation phenomena related to Bose-Einstein particles is analyzed. The model is a Fokker-Planck-type approximation of the Boltzmann-Nordheim equation, only keeping the leading order…
It is well-known that small, regular, spherically symmetric characteristic initial data to the Einstein-scalar-field system which are decaying towards (future null) infinity give rise to solutions which are foward-in-time global (in the…