Related papers: Some remarks on the continuity equation
We solve the Cauchy problem of the Ward equation with both continuous and discrete scattering data.
We consider the Cauchy problem for Schr\"odinger type operators. Under a suitable decay assumption on the imaginary part of the first order coefficients we prove well-posedness of the Cauchy problem in Gelfand-Shilov classes. We also…
Here we have introduced the idea of rough Cauchyness of sequences in a cone metric space. Also here we have discussed several basic properties of rough Cauchy sequences in a cone metric space using the idea of Phu.
The exact solution of a Cauchy problem related to a linear second-order difference equation with constant noncommutative coefficients is reported.
Initial value problems for the integrable discrete equations on quad-graphs are investigated. A geometric criterion of the well-posedness of such a problem is found. The effects of the interaction of the solutions with the localized defects…
An atomic random complex measure defined on the unit disk with Normally distributed moments is considered. An approximation to the distribution of the zeros of its Cauchy transform is computed. Implications of this result for solving…
Parabolic integro-differential Kolmogorov equations with different space-dependent operators are considered in H\"{o}lder-type spaces defined by a scalable L\'{e}vy measure. Probabilistic representations are used to prove continuity of the…
For a one-dimensional mildly quasilinear wave equation given in the upper half-plane, we consider the Cauchy problem. The initial conditions have discontinuity of the first kind at one point. We construct the solution using the method of…
In this work we shall study the well-posedness and ill-posedness of the Cauchy problem associated to the equation \begin{equation*} u_{t}+a(u^{n})_{x}+(b\mathscr{H} u_{t}+u_{yy})_{x}=0, \end{equation*} in anisotropic weigthed Sobolev…
Some Ostrowski type inequalities via Cauchy's mean value theorem and applications for certain particular instances of functions are given.
We consider the Cauchy problem for wave equations with unbounded damping coefficients in the whole space. For a general class of unbounded damping coefficients, we derive uniform total energy decay estimates together with a unique existence…
The solutions of Rashevskii equation for gonometric family of plane curves are considered. Their properties are studied. The connection with the theory of duality for the second order ODE's is discussed.
We prove weighted estimates for the maximal regularity operator. Such estimates were motivated by boundary value problems. We take this opportunity to study a class of weak solutions to the abstract Cauchy problem. We also give a new proof…
We demonstrate a quantitative version of the usual properties related to unique continuation from an interior datum for the Schr\"odinger equation with bounded or unbounded potential. The inequalities we establish have constants that…
In this paper we study the continuous dependence with respect to obstacles for obstacle problems with measure data. This is deeply investigated introducing a suitable type of convergence, which gives stability under very general hypotheses.…
In this paper, we study the Cauchy problem for a wave equation with general strong damping $-\mu(|D|)\Delta u_t$ motivated by [Tao, Anal. PDE (2009)] and [Ebert-Girardi-Reissig, Math. Ann. (2020)]. By employing energy methods in the Fourier…
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…
We extend Painlev\'e's determinateness theorem from the theory of ordinary differential equations in the complex domain allowing more general 'multiple-valued' Cauchy's problems. We study $C^0-$continuability (near singularities) of…
It is well known that Cauchy problem for Laplace equations is an ill-posed problem in Hadamard's sense. Small deviations in Cauchy data may lead to large errors in the solutions. It is observed that if a bound is imposed on the solution,…
This paper investigates the nonlinear Schr\"{o}dinger equation with a singular convolution potential. It demonstrates the local well-posedness of this equation in a modified Sobolev space linked to the energy. Additionally, we derive…