Related papers: Singularities of a characteristic Cauchy problem f…
We consider the Cauchy problem for a second order quasi-linear partial differential equation with an admissible parabolic degeneration such that the given functions described the initial conditions are defined on a closed interval. We study…
We consider the Cauchy problem for semilinear parabolic equation in divergence form with obstacle. We show that under natural conditions on the right-hand side of the eqution and mild conditions on the obstacle a unique continuous solution…
A comprehensive convergence and stability analysis of some probabilistic numerical methods designed to solve Cauchy-type inverse problems is performed in this study. Such inverse problems aim at solving an elliptic partial differential…
We study a class of hyperbolic Cauchy problems, associated with linear operators and systems with polynomially bounded coefficients, variable multiplicities and involutive characteristics, globally defined on R^n. We prove well-posedness in…
In this paper, we study the wave equation on infinite graphs. On one hand, in contrast to the wave equation on manifolds, we construct an example for the non-uniqueness for the Cauchy problem of the wave equation on graphs. On the other…
In this manuscript, we highlight a new phenomenon of complex algebraic singularity formation for solutions of a large class of genuinely nonlinear partial differential equations (PDEs). We start from a unique Cauchy datum, which is…
We study the problem of existence, uniqueness and regularity of probabilistic solutions of the Cauchy problem for nonlinear stochastic partial differential equations involving operators corresponding to regular (nonsymmetric) Dirichlet…
This paper is devoted to strictly hyperbolic systems and equations with non-smooth coefficients. Below a certain level of smoothness, distributional solutions may fail to exist. We construct generalised solutions in the Colombeau algebra of…
The paper provides a direct proof the uniqueness of solutions to the Camassa-Holm equation, based on characteristics. Given a conservative solution $u=u(t,x)$, an equation is introduced which singles out a unique characteristic curve…
We study a generalization of constant Gauss curvature -1 surfaces in Euclidean 3-space, based on Lorentzian harmonic maps, that we call pseudospherical frontals. We analyze the singularities of these surfaces, dividing them into those of…
A general class of singular abstract Cauchy problems is considered which naturally arises in applications to certain Free Boundary Problems. Existence of an associated evolution operator characterizing its solutions is established and is…
We describe some relations between the properties of the Cauchy problem for an ODE and the properties of the Cauchy problem for the associated continuity equation in the class of measures.
We construct a sequence that converges to a solution of the Cauchy problem for a singularly perturbed linear inhomogeneous differential equation of an arbitrary order. This sequence is also an asymptotic sequence in the following sense: the…
In this paper we investigate the Cauchy problem for Schr\"odinger ultrahyperbolic equations with singular (less than continuous) coefficients. We prove $H^\infty$ well-posedness in the very weak sense under suitable assumptions of the…
We study the Cauchy problem for a scalar semilinear degenerate parabolic partial differential equation with stochastic forcing. In particular, we are concerned with the well-posedness in any space dimension. We adapt the notion of kinetic…
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…
In this paper, using the similarity method, we construct particular solutions with singularities for degenerate high-order equations. The considered equations have singularities of the first and second kind. Particular solutions are…
We study the Cauchy problem for a general homogeneous linear partial differential equation in two complex variables with constant coefficients and with divergent initial data. We state necessary and sufficient conditions for the summability…
An example of a solution branch of the three dimensional Euler equation Cauchy problem is constructed which develops a singular velocity component and a singular vorticity component after finite time for some data which have Hoelder…
This paper is devoted to the investigation of propagation of singularities in hyperbolic equations with non-smooth oefficients, using the Colombeau theory of generalized functions. As a model problem, we study the Cauchy problem for the…