Related papers: An example of global classical solution for the Pe…

In this paper, we study the Cauchy problem for a generalized integrable Camassa-Holm equation with both quadratic and cubic nonlinearity. By overcoming the difficulties caused by the complicated mixed nonlinear structure, we firstly…

We establish local and global well-posedness for the Cauchy problem of a generalized Camassa-Holm equation where orders of the momentum and the nonlinearity can be arbitrarily high. More precisely, we consider the equation \begin{equation*}…

Complex metrics are a double-edged sword: they allow one to replace singular spacetimes, such as those containing a big bang, with regular metrics, yet they can also describe unphysical solutions in which quantum transitions may be more…

We consider the solutions of the Cauchy problem for a dyadic model of Euler equations. We prove global existence and uniqueness of Leray-Hopf solutions in a rather large class K that implies in particular global existence and uniqueness in…

The author gives an alternative and simple proof of the global existence of smooth solutions to the Cauchy problem for wave maps from the 1+2-dimensional Minkowski space to an arbitrary compact smooth Riemannian manifold without boundary,…

We derive global analytic representations of fundamental solutions for a class of linear parabolic systems with full coupling of first order derivative terms where coefficient may depend on space and time. Pointwise convergence of the…

We investigate the Cauchy problem for a two-component generalization of the Novikov equation with cubic nonlinearity -- an integrable system whose solutions may develop strong nonlinear phenomena such as gradient blow-up and interactions…

Conditions for the unique solvability of the Cauchy problem for a family of scalar functional differential equations are obtained. These conditions are sufficient for the solvability of the Cauchy problem for every equation from the family…

Let $(u,v)$ be a solution to the Cauchy problem for a semilinear parabolic system \[ \mathrm{(P)} \qquad \cases{ \partial_t u=D_1\Delta u+v^p\quad & $\quad\mbox{in}\quad{\mathbb{R}}^N\times(0,T),$\\ \partial_t v=D_2\Delta v+u^q\quad &…

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…

In this paper, we verify the Glassey conjecture in the radial case for all spatial dimensions, which states that, for the nonlinear wave equations of the form $\Box u=|\nabla u|^p$, the critical exponent to admit global small solutions is…

We prove that, for first-order, fully nonlinear systems of partial differential equations, under an hypothesis of ellipticity for the principal symbol, the Cauchy problem has no solution within a range of Sobolev indices depending on the…

We discuss the application of recent results on generalized solutions to the Cauchy problem for hyperbolic systems to Dirac equations with external fields. In further analysis we focus on the question of existence of associated…

In recent years, the global existence of classical solutions to the Cauchy problem for 2D incompressible viscous MHD equations without magnetic diffusion has been proved in \cite{Ren,TZhang}, under the assumption that initial data is close…

In this paper we study the solvability of the Cauchy-Riemann equation with prescibed support in different spaces of forms. The unbounded Hartogs triangle in $\mathbb C^2$ and the Hartogs domains in $\mathbb C\mathbb P^2$ provide us new…

We study non-linear evolution equations with periodic initial conditions. In particular, we use the graph method introduced by Galavotti to prove the existence of global solution of Hamiltonian perturbation of KdV without any restriction on…

We consider the following chemotaxis system under homogeneous Neumann boundary conditions in a smooth, open, bounded domain $\Omega \subset \mathbb{R}^n$ with $n \geq 3$: \begin{equation*} \begin{cases} u_t = \Delta u - \chi \nabla \cdot…

We prove that solutions to the critical wave equation below can not be global if the initial values are positive somewhere and nonnegative. This completes the solution to the famous blow up conjecture about critical semilinear wave…

In this paper, we establish the existence of a 1-parameter family of spatially inhomogeneous radially symmetric classical self-similar solutions to a Cauchy problem for a semi-linear parabolic PDE with non-Lipschitz nonlinearity and trivial…

We consider the Cauchy problem for linearly damped nonlinear Schr\"odinger equations \[ i\partial_t u + \Delta u + i a u= \pm |u|^\alpha u, \quad (t,x) \in [0,\infty) \times \mathbb{R}^N, \] where $a>0$ and $\alpha>0$. We prove the global…