Related papers: Norm inflation for the viscous nonlinear wave equa…

In this paper, we consider the existence of positive solutions with prescribed $L^2$-norm for the following nonlinear Schr\"{o}dinger equation involving potential and Sobolev critical exponent \begin{equation*} \begin{cases} -\Delta…

We prove that the Cauchy problem for the three dimensional Navier-Stokes equations is ill posed in $\dot{B}^{-1,\infty}_{\infty}$ in the sense that a ``norm inflation'' happens in finite time. More precisely, we show that initial data in…

In this work we study the blow-up of solutions of a weakly coupled system of damped semilinear wave equations in the scattering case with power nonlinearities. We apply an iteration method to study both the subcritical case and the critical…

New local smoothing estimates in Besov spaces adapted to the half-wave group are proved via $\ell^2$-decoupling. We apply these estimates to obtain new well-posedness results for the cubic nonlinear wave equation in two dimensions. The…

We describe regularizing effects in the linearization of a kinetic equation that arises in study of a system of nonlinear waves satisfying the Schr\"odinger equation in terms of weak turbulence and condensate. The problem is first…

In this paper, we prove the existence and regularity of weak positive solutions for a class of nonlinear elliptic equations with a singular nonlinearity, lower order terms and $L^{1}$ datum in the setting of variable exponent Sobolev…

In this paper, we consider an inverse problem for a nonlinear wave equation with a damping term and a general nonlinear term. This problem arises in nonlinear acoustic imaging and has applications in medical imaging and other fields. The…

We consider the semi-classical limit for the Gross-Pitaevskii equation. In order to consider non-trivial boundary conditions at infinity, we work in Zhidkov spaces rather than in Sobolev spaces. For the usual cubic nonlinearity, we obtain a…

We study the three-dimensional cubic nonlinear wave equation (NLW) with random initial data below $L^2(\mathbb{T}^3)$. By considering the second order expansion in terms of the random linear solution, we prove almost sure local…

We consider the Cauchy problem for the nonlinear wave equation $u_{tt} - \Delta_x u +q(t, x) u + u^3 = 0$ with smooth potential $q(t, x) \geq 0$ having compact support with respect to $x$. The linear equation without the nonlinear term…

In this paper, we consider the semilinear wave equation involving the nonlinear damping term $g(u_t) $ and nonlinearity $f(u)$. The well-posedness of the weak solution satisfying some additional regularity is achieved under the wider ranges…

In this paper, we obtain a series of regularity results for viscosity solutions of fully nonlinear elliptic equations with oblique derivative boundary conditions. In particular, we derive the pointwise $C^{\alpha}$, $C^{1,\alpha}$ and…

In this paper, we prove that the Cauchy problem for the Fornberg-Whitham equation is not locally well-posed in $B^s_{p,r}(\R)$ with $(s,p,r)\in (1,1+\frac1p)\times[2,\infty)\times [1,\infty]$ or $(s,p,r)\in \{1\}\times[2,\infty)\times…

We prove global stability for a system of nonlinear wave equations satisfying a generalized null condition. The generalized null condition allows for null forms whose coefficients have bounded $C^k$ norms. We prove both pointwise decay and…

This paper finds solutions to semilinear wave equations with strongly anomalous propagation of singularities. For very low Sobolev regularity we obtain solutions whose singular support propagates along any ray inside or outside the light…

We present a method by which cosmological perturbations can be quantitatively studied in single and multi-field inflationary models beyond linear perturbation theory. A non-linear generalization of the gauge-invariant Sasaki-Mukhanov…

We consider the \emph{focusing} nonlinear Schr\"odinger equation posed on the one dimensional line, with nonzero background condition at spatial infinity, given by a homogeneous plane wave. For this problem of physical interest, we study…

The object of this study is to investigate the effect of viscosity on propagation of free-surface waves in an incompressible viscous fluid layer of arbitrary depth. While we provide a more detailed study of properties of linear surface…

We study the nonlinear wave equation with a sign-changing potential in any space dimension. If the potential is small and rapidly decaying, then the existence of small-amplitude solutions is driven by the nonlinear term. If the potential…

We prove nonlinear modulational instability for both periodic and localized perturbations of periodic traveling waves for several dispersive PDEs, including the KDV type equations (e.g. the Whitham equation, the generalized KDV equation,…