Related papers: Global Weak Solutions for the Half-Wave Maps Equat…
We study the two-dimensional wave equation with cubic nonlinearity posed on $\mathbb R^2$, with space-time white noise forcing. After a suitable renormalisation of the nonlinearity, we prove global well-posedness for this equation for…
We investigate the semilinear wave equation with potential on weighted graphs. We establish sufficient conditions for the nonexistence of global-in-time solutions. Both nonnegative and sign-changing solutions are considered. In particular,…
In this work we study weighted total least squares problems on infinite dimensional spaces. We show that in most cases this problem does not admit a solution (except in the trivial case) and then, we consider a regularization on the…
We prove the global well-posedness of weak solutions for nonlinear wave equations with supercritical source and damping terms on a three-dimensional torus $\mathbb T^3$ of the prototype \begin{align*} &u_{tt}-\Delta…
In this paper, we show almost global existence of small solutions to the Cauchy problem for symmetric system of wave equations with quadratic (in 3D) or cubic (in 2D) nonlinear terms and multiple propagation speeds. To measure the size of…
In this paper, we prove that the existence and uniqueness of globally weak solutions to the Cauchy problem for the weakly dissipative Camassa-Holm equation in time weighted $H^1$ space. First, we derive an equivalent semi-linear system by…
We study a system of semilinear wave equations on Kerr backgrounds that satisfies the weak null condition. Under the assumption of small initial data, we prove global existence and pointwise decay estimates.
We define the weak-normalization and the seminormalization of a real algebraic variety relative to its central locus. The study is related to the properties of the rings of continuous rational functions and hereditarily rational functions…
We study the relations between different regularity assumptions in the definition of weak solutions and supersolutions to the porous medium equation. In particular, we establish the equivalence of the conditions $u^m \in L^2_{\rm…
In a recent article, we studied the global solvability of the so-called cohomological equation L_X f=g in C^\infty(\Rt), where X is a regular vector field on the plane and L_X the corresponding Lie derivative. In a joint article with T.…
We prove uniqueness of solutions to the wave map equation in the natural class, namely $ (u, \partial_t u) \in C([0,T); \dot{H}^{d/2})\times C^1([0,T); \dot{H}^{d/2-1})$ in dimensions $d\geq 4$. This is achieved through estimating the…
We establish the global existence of weak solutions of the isentropic compressible magnetohydrodynamic equations with ripped density in the whole plane provided the bulk viscosity coefficient is properly large. Moreover, we show that such…
The purpose of this note is to prove the existence of global weak solutions to the flow associated to integro-differential harmonic maps into spheres and Riemannian homogeneous manifolds.
In this paper, we study the global existence of solutions of the Cauchy problem for a class of weakly dissipative nonlinear dispersive wave equations…
In this paper, we study ill-posedness of cubic fractional nonlinear Schr\"odinger equations. First, we consider the cubic nonlinear half-wave equation (NHW) on $\mathbb R$. In particular, we prove the following ill-posedness results: (i)…
This study investigates a semilinear wave equation characterized by nonlinear damping $g(u_t) $ and nonlinearity $f(u)$. First, the well-posedness of weak solutions across broader exponent ranges for $g$ and $f$ is established, by utilizing…
In the paper [H. Kubo, Global existence for exterior problems of semilinear wave equations with the null condition in 2D, Evol. Equ. Control Theory 2 (2013), no. 2, 319-335], for the 2-D semilinear wave equation system…
We consider the initial-value problem for stochastic continuity equations of the form $$ \partial_t \rho + \text{div}_h \left[\rho \left(u(t,x) + \sum_{i=1}^N a_i(x)\circ \frac{dW^i}{dt}\right)\right] = 0, $$ defined on a smooth closed…
We show global existence backwards from scattering data at infinity for semilinear wave equations satisfying the null condition or the weak null condition. Semilinear terms satisfying the weak null condition appear in many equations in…
Half-wave maps appear in the physics literature as the continuum limit of Calogero-Moser spin systems. We obtain a uniqueness result for the Half-Wave Maps equation in dimension $d \ge 3$ in the natural energy class with $\mathbb{H}^2$…