Related papers: Wave decay on convex co-compact hyperbolic manifol…
On a compact manifold $M^{n}$ ($n\geq 3$) with boundary, we study the asymptotic behavior as $\epsilon$ tends to zero of solutions $u_{\epsilon}: M \to \mathbb{C}$ to the equation $\Delta u_{\epsilon} + \epsilon^{-2}(1 -…
In this note, we study weak solutions of equation \begin{equation}\label{eq 00.1} \Delta u =\frac{4e^u}{1+e^u} -4\pi\sum^{N}_{i=1}\delta_{p_i}+4\pi\sum^{M}_{j=1}\delta_{q_j} \quad{\rm in}\;\; \mathbb{R}^2, \end{equation} where…
We consider a hyperbolic system of conservation laws u_t + f(u)_x = 0 and u(0,\cdot) = u_0, where each characteristic field is either linearly degenerate or genuinely nonlinear. Under the assumption of coinciding shock and rarefaction…
We obtain a vanishing result for solutions of the inequality $|\Delta u|\le q_1|u|+q_2|\nabla u|$ that decay to zero along a very general warped cylindrical end of a Riemannian manifold. The appropriate decay condition at infinity on $u$ is…
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…
We prove local decay estimates for the wave equation in the asymptotically Euclidean setting. In even dimensions we go beyond the optimal decay by providing the large time asymptotic profile, given by a solution of the free wave equation.…
We study the large time behavior of solutions to the porous medium equation in nonhomogeneous media with critical singular density $$ |x|^{-2}\partial_{t}u=\Delta u^m, \quad \hbox{in} \ \real^N\times(0,\infty), $$ where $m>1$ and $N\geq3$.…
In the paper, we are concerned with the large time asymptotics toward the viscous contact waves for solutions of the Landau equation with physically realistic Coulomb interactions. Precisely, for the corresponding Cauchy problem in the…
We study the regularity of weak solutions to a certain class of second order parabolic system under the only assumption of continuous coefficients. By using the $A-$caloric approximation argument, we claim that the weak solution $u$ to such…
We show that, given any static spacetime whose spatial slices are asymptotically Euclidean (or, more generally, asymptotically conic) manifolds modeled on the large end of the Schwarzschild exterior, there exist stationary solutions to the…
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…
This paper is concerned with the study of a nonlinear non-local equation that has a commutator structure. The equation reads $\partial_t u-F(u) (-\Delta)^{s/2} u+(-\Delta)^{s/2} (uF(u))=0$, $x\in \mathbb{T}^d$, with s $\in$ (0, 1]. We are…
We study the asymptotic behavior of large data solutions to Schr\"odinger equations $i u_t + \Delta u = F(u)$ in $\R^d$, assuming globally bounded $H^1_x(\R^d)$ norm (i.e. no blowup in the energy space), in high dimensions $d \geq 5$ and…
Goal of this paper is to study the asymptotic behaviour of the solutions of the following doubly nonlocal equation $$(-\Delta)^s u + \mu u = (I_{\alpha}*F(u))f(u) \quad \hbox{on $\mathbb{R}^N$}$$ where $s \in (0,1)$, $N\geq 2$, $\alpha \in…
In the first part of this paper, the existence of infinitely many $L^p$-standing wave solutions for the nonlinear Helmholtz equation $$ -\Delta u -\lambda u=Q(x)|u|^{p-2}u\quad\text{ in }\mathbb{R}^N $$ is proven for $N\geq 2$ and…
For the three-dimensional vacuum free boundary problem with physical singularity that the sound speed is $C^{ {1}/{2}}$-H$\ddot{\rm o}$lder continuous across the vacuum boundary of the compressible Euler equations with damping, without any…
We show that for a very general class of curvature functions defined in the positive cone, the problem of finding a complete strictly locally convex hypersurface in $H^n+1$ satisfying $f(\kappa)=\sigma\in(0, 1)$ with a prescribed asymptotic…
We consider the hyperbolic-parabolic singular perturbation problem for a degenerate quasilinear Kirchhoff equation with weak dissipation. This means that the coefficient of the dissipative term tends to zero when t tends to +infinity. We…
Partial wave expansion of the Coulomb-distorted plane wave is determined and studied. Dominant and sub-dominant asymptotic expansion terms are given and leading order three-dimensional asymptotic form is derived. The generalized…
We establish the decay of the solutions of the damped wave equations in one dimensional space for the Dirichlet, Neumann, and dynamic boundary conditions where the damping coefficient is a function of space and time. The analysis is based…