Related papers: Global Well-Posedness of a Class of Strictly Hyper…

By a further investigation on the structure of the coefficient matrix of the globally hyperbolic regularized moment equations for Boltzmann equation in [Z. Cai, Y. Fan and R. Li, Comm. Math. Sci., 11 (2013), pp. 547-571], we propose a…

We consider several classes of degenerate hyperbolic equations involving delay terms and suitable nonlinearities. The idea is to rewrite the problems in an abstract way and, using semigroup theory and energy method, we study well posedness…

This paper studies the Cauchy problem for variable coefficient weakly hyperbolic first order systems of partial differential operators. The hyperbolicity assumption is that for each $t, x$ the principal symbol is hyperbolic. No hypothesis…

The present paper concerns the well-posedness of the Cauchy problem for microlocally symmetrizable hyperbolic systems whose coefficients and symmetrizer are log-Lipschitz continuous, uniformly in time and space variables. For the global in…

In this paper we present a method to study global regularity properties of solutions of large-data critical Schrodinger equations on certain noncompact Riemannian manifolds. We rely on concentration compactness arguments and a global…

In this paper we study the Cauchy problem associated to the Maxwell-Schr\"odinger system with a defocusing pure-power non-linearity. This system has many applications in physics, for instance in the description of a charged non-relativistic…

In this paper, we consider the Cauchy problem for a hyperbolic equation $Q(\partial_t,\partial_x)u=0$ of any order $m\geq3$, where $t\geq0$ and $x\in\mathbb{R}^n$, and $Q=P_m+P_{m-1}+P_{m-2}$ is a sum of homogeneous hyperbolic polynomials…

In this paper we prove global existence of weak solutions, their regularization, and relaxation for large data for a broad class of Fokker-Planck-Alignment models which appear in collective dynamics. The main feature of these results, as…

In this paper we consider weakly hyperbolic equations of higher orders in arbitrary dimensions with time-dependent coefficients and lower order terms. We prove the Gevrey well-posedness of the Cauchy problem under $C^k$-regularity of…

In this paper we consider the Cauchy problem for higher order weakly hyperbolic equations. We assume that the principal symbol depends only on one space variable and the characteristic roots $\tau_j$ verify the inequality \[\tau_j^2(x) +…

In this paper we study higher order weakly hyperbolic equations with time dependent non-regular coefficients. The non-regularity here means less than H\"older, namely bounded coefficients. As for second order equations in \cite{GR:14} we…

In the present paper, we consider second order strictly hyperbolic linear operators of the form $Lu\,=\,\partial_t^2u\,-\,{\rm div}\big(A(t,x)\nabla u\big)$, for $(t,x)\in[0,T]\times\mathbb{R}^n$. We assume the coefficients of the matrix…

This paper considers hyperbolic wave equations with non-local in time conditions involving integrals with respect to time. It is shown that regularity of the solution can be achieved for complexified problem with integral conditions…

This paper studies the well-posedness of a class of nonlocal parabolic partial differential equations (PDEs), or equivalently equilibrium Hamilton-Jacobi-Bellman equations, which has a strong tie with the characterization of the equilibrium…

We consider the relativistic, spatially inhomogeneous Fokker-Planck equation with an external confining potential. We prove the exponential time decay of solutions towards the global equilibrium in weighted $L^2$ and Sobolov spaces. Our…

We consider the Cauchy problem for a hyperbolic pseudodifferential operator whose symbol is generalized, resembling a representative of a Colombeau generalized function. Such equations arise, for example, after a reduction-decoupling of…

We study a class of parabolic equations having first order terms with superlinear (and subquadratic) growth. The model problem is the so-called viscous Hamilton-Jacobi equation with superlinear Hamiltonian. We address the problem of having…

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…

We study first-order symmetrizable hyperbolic $N\times N$ systems in a spacetime cylinder whose lateral boundary is totally characteristic. In local coordinates near the boundary at $x=0$, these systems take the form \[ \partial_t u +…

We consider the Cauchy problem of the viscous $\beta$-plane equations. We first establish the global well-posedness of the system for the initial data sufficiently small compared to the Rossby parameter. The smoothing effect of the flow is…