Related papers: The periodic Cauchy problem of the modified Hunter…
We study, in the periodic setting, the well-posedness of the Cauchy problem associated to the operator $P(t, D_{x}, D_{t}) = D_{t} - a_{2}(t) \Delta_{x} + \sum_{j = 1}^{N} a_{1, j}(t) D_{x_{j}} + a_{0}(t)$, with $T> 0$, $t \in [0, T]$ and…
We consider the Cauchy problem for the Kirchhoff equation on $\mathbb{T}^d$ with initial data of small amplitude $\varepsilon$ in Sobolev class. We prove a lower bound $\varepsilon^{-4}$ for the existence time, which improves the bound…
In the paper, a linear differential equation with variable coefficients and a Caputo fractional derivative is considered. For this equation, a Cauchy problem is studied, when an initial condition is given at an intermediate point that does…
The aim of this paper is to investigate well-posedness of the Cauchy problem for the degenerate Zakharov system. Local well-posedness holds for anisotropic Sobolev data by applying $U^2, V^2$ type spaces. We give the Schr\"odinger initial…
In this article we consider the Cauchy problem with large initial data for an equation of the form (\partial_t+\partial_x^3)u=F(u,u_x,u_{xx}) where F is a polynomial with no constant or linear terms. Local well-posedness was established in…
In this paper, we investigate an ill-posed Cauchy problem involving a stochastic parabolic equation. We first establish a Carleman estimate for this equation. Leveraging this estimate, we derive the conditional stability and convergence…
We study the initial value problem in Einstein-Cartan theory which includes torsion and, therefore, a non-symmetric connection on the spacetime manifold. Generalizing the path of a classical theorem by Choquet-Bruhat and York for the…
The following stochastic Cauchy initial-value problem is studied for the parabolic heat equation on a domain $ \mathbf{Q}\subset{\mathbf{R}}^{n}$ with random field initial data. \begin{align} &{\square}\widehat{u(x,t)} \equiv…
In this paper, we study the Cauchy problem for the linear spatially homogeneous Boltzamnn equation with Debye-Yukawa potential. Using the spectral decomposition of the linear operator, we prove that, for an initial datum in the sense of…
We consider the Cauchy problem for an equation of the form \partial_t+\partial_x^3)u=F(u,u_x,u_{xx}) where F is a polynomial with no constant or linear terms and no quadratic uu_{xx} term. For a polynomial nonlinearity with no quadratic…
We prove the existence of a fundamental solution of the Cauchy initial boundary value problem on the whole space for a parabolic partial differential equation with discontinuous unbounded first-order coefficient at the origin. We establish…
We study the Cauchy problem for the semilinear heat equation with the singular potential, called the Hardy-Sobolev parabolic equation, in the energy space. The aim of this paper is to determine a necessary and sufficient condition on…
We prove a local in time well-posedness result for quasi-linear Hamiltonian Schr\"odinger equations on $\mathbb{T}^d$ for any $d\geq 1$. For any initial condition in the Sobolev space $H^s$, with $s$ large, we prove the existence and…
We consider two different relativistic versions of the Kepler problem in the plane: the first one involves the relativistic differential operator, the second one involves a correction for the usual gravitational potential due to…
We consider primal-dual mixed finite element methods for the solution of the elliptic Cauchy problem, or other related data assimilation problems. The method has a local conservation property. We derive a priori error estimates using known…
In this paper, we consider a Cauchy problem for a first-order hyperbolic equation with time-dependent coefficients. Cauchy data are given on a lateral subboundary and we obtain local H\"older stabilities for inverse source and coefficient…
We derive an initial value formulation for dynamical Chern-Simons gravity, a modification of general relativity involving parity-violating higher derivative terms. We investigate the structure of the resulting system of partial differential…
A classical problem in general relativity is the Cauchy problem for the linearised Einstein equation (the initial value problem for gravitational waves) on a globally hyperbolic vacuum spacetime. A well-known result is that it is uniquely…
We study the large time behavior of solutions of first-order convex Hamilton-Jacobi Equations of Eikonal type set in the whole space. We assume that the solutions may have arbitrary growth. A complete study of the structure of solutions of…
The Cauchy problem for the Camassa -- Holm equation with step-like initial conditions is reformulated as a Riemann -- Hilbert problem. The the initial value problem solution is obtained then in a parametric form from the Riemann -- Hilbert…