Related papers: Global classical solutions to partially dissipativ…

For first-order quasi-linear systems of partial differential equations, we formulate an assumption of a transition from initial hyperbolicity to ellipticity. This assumption bears on the principal symbol of the first-order operator. Under…

We investigate the Cauchy problem for a semilinear parabolic equation driven by a mixed local-nonlocal diffusion operator of the form \[ \partial_t u - (\Delta - (-\Delta)^{\mathsf{s}})u = \mathsf{h}(t)|x|^{-b}|u|^p + t^\varrho…

As a continuation of the previous work [40], in this paper we focus on the Cauchy problem of the two-dimensional (2D) incompressible Boussinesq equations with fractional Laplacian dissipation. We give an elementary proof of the global…

The paper studies some ill-posed boundary value problems on semi-plane for parabolic equations with homogenuous Cauchy condition at initial time and with the second order Cauchy condition on the boundary of the semi-plane. A class of inputs…

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…

We prove that for large enough data, the life span of smooth solutions to the Cauchy problem for the following two quasilinear hyperbolic systems is finite: (1) equations of relativistic compressible fluid dynamics, (2) equations of plasma…

We prove existence of a global conservative solution of the Cauchy problem for the two-component Camassa-Holm (2CH) system on the line, allowing for nonvanishing and distinct asymptotics at plus and minus infinity. The solution is proven to…

We study the quasilinear non-local Benney System $$\left\{\begin{array}{llll} iu_t+u_{xx}=|u|^2u+buv\\ v_t+a(\int_{\mathbf{R}^+}v^2dx)v_x=-b(|u|^2)_x,\quad (x,t)\in\mathbf{R}^+\times [0,T],\, T>0. \end{array}\right.$$ We establish the…

We prove local existence and uniqueness of the Cauchy problem for a large class of tensorial second order linear hyperbolic partial differential equations with coefficients of low regularity in a suitable class of generalized functions.

In this paper we consider some new classes of integral equations that arise from Lie symmetry analysis. Specifically, we consider the task of obtaining solutions of a Cauchy problem for some classes of second order hyperbolic partial…

We introduce novel approximate systems for dispersive and diffusive-dispersive equations with nonlinear fluxes. For purely dispersive equations, we construct a first-order, strictly hyperbolic approximation. Local well-posedness of smooth…

We consider the Cauchy problem for the nonlinear dynamical Lam\'e system with double wave speeds in a $d$-dimensional $(d=2,3)$ periodic domain. Moreover, the equations can be transformed into a linearly degenerate hyperbolic system. We…

The Einstein evolution equations are studied in a gauge given by a combination of the constant mean curvature and spatial harmonic coordinate conditions. This leads to a coupled quasilinear elliptic--hyperbolic system of evolution…

In this paper, we investigate the asymptotic behavior of solutions to the Cauchy problem for the scalar viscous conservation law where the far field states are prescribed. Especially, we deal with the case when the viscosity is of…

We consider the Cauchy problem for the wave equation on a non-globally hyperbolic manifold of the special form (Minkowski plane with a handle) containing closed timelike curves (time machines). We prove that the classical solution of the…

In this paper, we study the ill-posdness of the Cauchy problem for semilinear wave equation with very low regularity, where the nonlinear term depends on $u$ and $\partial_t u$. We prove a ill-posedness result for the "defocusing" case, and…

We study the Cauchy problem for a system of cubic nonlinear Klein-Gordon equations in one space dimension. Under a suitable structural condition on the nonlinearity, we will show that the solution exists globally and decays of the order…

Given a strictly hyperbolic $n\times n$ system of conservation laws, it is well known that there exists a unique Lipschitz semigroup of weak solutions, defined on a domain of functions with small total variation, which are limits of…

We study the Cauchy problem for fully nonlinear (stochastic) parabolic partial differential equations. We provide both in deterministic and stochastic case the existence of a maximal defined solution for the problem and we provide suitable…

For a certain class of genuinely nonlinear two-by-two planar hyperbolic systems we show that any classical solution on a smoothly bounded domain has nontangential boundary limits except on a set whose Hausdorff dimension is bounded by some…