Related papers: On the Cauchy problem in Sobolev spaces for nonlin…
We show the short time existence and uniqueness of solutions to the Cauchy problem for fully nonlinear systems of arbitrary even order on closed manifolds which are strongly parabolic at the initial values. The proof uses a linearization…
We consider the Schr{\"o}dinger equation with a logarithmic nonlinearty and non-trivial boundary conditions at infinity. We prove that the Cauchy problem is globally well posed in the energy space, which turns out to correspond to the…
We consider the nonlinear Schr\"odinger equations with a potential on $\mathbb T^d$. For almost all potentials, we show the almost global stability in very high Sobolev norms. We apply an iteration of the Birkhoff normal form, as in the…
We carry out an analysis of the existence of solutions for a class of nonlinear partial differential equations of parabolic type. The equation is associated to a nonlocal initial condition, written in general form which includes, as…
The spatially periodic initial problem and Cauchy problem for nonlinear Schr\"odinger equations are considered. The existence and uniqueness of global solution with infinite smooth initial data $u_0$, i.e. $u_0,\;|u_0|^{2p}u_0\in…
In this paper, we study the Cauchy problem for the energy-critical inhomogeneous nonlinear Schr\"{o}dinger equation with inverse-square potential \[iu_{t} +\Delta u-c|x|^{-2}u=\lambda|x|^{-b} |u|^{\sigma } u,\; u(0)=u_{0} \in…
We show that hyperbolicity is a necessary condition for the well posedness of the noncharacteristic Cauchy problem for nonlinear partial differential equations. We give conditions on the initial data which are necessary for the existence of…
We study the Cauchy problem for the cubic nonlinear Schroedinger equation, perturbed by (higher order) dissipative nonlinearities. We prove global in-time existence of solutions for general initial data in the energy space. In particular we…
In this paper, we investigate the problems of Cauchy theory and exponential stability for the inhomogeneous Boltzmann equation without angular cut-off. We only deal with the physical case of hard potentials type interactions (with a…
We consider the Cauchy problem for strictly hyperbolic $m$-th order partial differential equations with coefficients low-regular in time and smooth in space. It is well-known that the problem is $L^2$ well-posed in the case of Lipschitz…
We study necessary conditions and sufficient conditions for the existence of local-in-time solutions of the Cauchy problem for superlinear fractional parabolic equations. Our conditions are sharp and clarify the relationship between the…
We consider the Cauchy problem of a dissipative nonlinear Schr\"odinger equation with a time dependent harmonic potential. We find a critical situation that the $L^2$-norm of dissipative solutions decays or not and which is decided by a…
We consider the Cauchy problem in the Euclidean space for a doubly degenerate parabolic equation with a space-dependent exponential weight, roughly speaking of the type of the exponential of a power of the distance from the origin. We…
In this work we consider a system of nonlinear Schr\"odinger equations whose nonlinearities satisfy a power-type-growth. First, we prove that the Cauchy problem is local and global well-posedness in $L^2$ and $H^1$. Next, we establish the…
We show the strong convergence in arbitrary Sobolev norms of solutions of the discrete nonlinear Schr{\"o}dinger on an infinite lattice towards those of the nonlinear Schr{\"o}dinger equation on the whole space. We restrict our attention to…
We consider a quasilinear parabolic Cauchy problem with spatial anisotropy of orthotropic type and study the spatial localization of solutions. Assuming the initial datum is localized with respect to a coordinate having slow diffusion rate,…
We consider the Cauchy problem to the general defocusing and focusing $p\times q$ matrix nonlinear Schr\"{o}dinger (NLS) equations with initial data allowing arbitrary-order poles and spectral singularities. By establishing the…
We construct bounded classical solutions of the Boltzmann equation in the whole space without specifying any limit behaviors at the spatial infinity and without assuming the smallness condition on initial data. More precisely, we show that…
We prove the existence of global solutions to the Cauchy problem for noncommutative nonlinear wave equations in arbitrary even spatial dimensions where the noncommutativity is only in the spatial directions. We find that for existence there…
We establish existence of positive non-decreasing radial solutions for a nonlocal nonlinear Neumann problem both in the ball and in the annulus. The nonlinearity that we consider is rather general, allowing for supercritical growth (in the…