Related papers: Singularities of a characteristic Cauchy problem f…
Parabolic integro-differential nondegenerate Cauchy problem is considered in the scale of L_{p} spaces of functions whose regularity is defined by a Levy measure with O-regulary varying radial profile. Existence and uniqueness of a solution…
In this paper, we study a class of strongly degenerate ultraparabolic equations with analytic coefficients. We demonstrate that the Cauchy problem exhibits an analytic smoothing effect. This means that, with an initial datum belonging to…
Results of investigation of the asymptotic behavior of solutions to the Cauchy problems for a quasi-linear parabolic equation with a small parameter at a higher derivative near singular points of limit solutions are presented. Interest to…
We investigate the quantitative unique continuation of solutions to higher order elliptic equations with singular coefficients. Quantitative unique continuation described by the vanishing order is a quantitative form of strong unique…
In this paper, we study the solvability of a Cauchy- Dirichlet problem for nonlinear parabolic equation with non standard growths and nonlocal terms. We show the existence of weak solutions of the considered problem under more general…
Motivated by models for biofilm growth, we consider Cauchy problems for quasilinear reaction diffusion equations where the diffusion coefficient has a porous medium type degeneracy as well as a singularity. We prove results on the…
We construct a formal asymptotic series expansion for a general solution of the Brans-Dicke equations with a fluid source near a sudden singularity. This solution contains eleven independent arbitrary functions of the spatial coordinates as…
We present a series of recent results on the well-posedness of very singular parabolic stochastic partial differential equations. These equations are such that the question of what it even means to be a solution is highly non-trivial. This…
The Cauchy problem is studied for the self-adjoint and non-self-adjoint Schroedinger equations. We first prove the existence and uniqueness of solutions in the weighted Sobolev spaces. Secondly we prove that if potentials are depending…
We introduce a discrete delayed exponential depending on sequence of matrices. This discrete matrix gives a representation of a solution to the Cauchy problem for a discrete linear system with pure delay with sequence of matrices. We…
In this paper we give a meaning to the nonlinear characteristic Cauchy problem for the Wave Equation in base form by replacing it by a family of non-characteristic problems in an appropriate algebra of generalized functions. We prove…
The purpose of this paper is to investigate the Cauchy problem for the Gross-Pitaevskii infinite linear hierarchy of equations on $\mathbb{R}^n,$ $n \geq 1.$ We prove local existence and uniqueness of solutions in certain Sobolev type…
These Notes are intended for graduate or undergraduate students who have familiarity with Lebesgue measure theory, partial differential equations, and functional analysis. The main topics covered in this work are the study of the Cauchy…
In this paper, we prove existence and uniqueness of measure solutions for the Cauchy problem associated to the (vectorial) continuity equation with a non-local flow. We also give a stability result with respect to various parameters.
Via Carleman estimates we prove uniqueness and continuous dependence results for lateral Cauchy problems for linear integro-differential parabolic equations without initial conditions. The additional information supplied prescribes the…
We study the singularity formation of strong solutions to the two-dimensional (2D) Cauchy problem of the non-baratropic compressible magnetohydrodynamic equations without heat conductivity. It is proved that the strong solution exists…
We consider the Cauchy problem associated with a general parabolic partial differential equation in $d$ dimensions. We find a family of closed-form asymptotic approximations for the unique classical solution of this equation as well as…
We establish the existence of solutions of the Cauchy problem for a higher-order semilinear parabolic equation by introducing a new majorizing kernel. We also study necessary conditions on the initial data for the existence of local-in-time…
We consider a family of linear singularly perturbed Cauchy problems which combines partial differential operators and linear fractional transforms. We construct a collection of holomorphic solutions on a full covering by sectors of a…
The Cauchy problem for the two dimensional compressible Euler equations with data in the Sobolev space $H^s(\mathbb R^2)$ is known to have a unique solution of the same Sobolev class for a short time, and the data-to-solution map is…