Alexander Anatolyevich Panin

On local solvability and blow-up of solutions of an abstract nonlinear Volterra integral equation

A theorem on noncontinuable solutions is proved for abstract Volterra integral equations with operator-valued kernels (continuous and polar). It is shown that if there is no global solvability, then the C-norm of the solution is unbounded but does not tend to infinity in general. An example of Volterra equations whose noncontinuable solutions are unbounded but not infinitely large is constructed. It is shown that the theorems on noncontinuable solutions of the Cauchy problem for abstract equations of the first and nth kind (with a linear leading part) are special cases of the theorems proved in this paper.

On the nonextendable solution and blow-up of the solution of the one-dimensional equation of ion-sound waves in a plasma

The initial boundary-value problem for the equation of ion-sound waves in a plasma is studied. A theorem on the nonextendable solution is proved. Sufficient conditions for the blow-up of the solution in finite time and the upper bound for the blow-up time are obtained using the method of test functions.

Dependence of the efficiency of a posteriori estimation of the solution accuracy of an elliptic boundary-value problem on the problem data and estimation algorithm parameters

Based on numerical experiments, we analyzed the relationship between the efficiency of the error estimation method proposed by S.I. Repin for approximate solutions of elliptical equations and the problem data and estimation algorithm parameters used.

Local solvability and solution blow-up of one-dimensional equations of the Yajima–Oikawa–Satsuma type

We consider one-dimensional equations of the type of the Yajima–Oikawa–Satsuma ion acoustic wave equation and prove the local solvability. Using the test function method, we obtain sufficient conditions for solution blow-up and estimate the blow-up time.

Blow-up of the solution of an inhomogeneous system of Sobolev-type equations

We consider a model system of two inhomogeneous nonlinear Sobolev-type equations of sixth order with second-order time derivative and prove the local (with respect to time) solvability of the problem. We state conditions under which the blow-up of the solution occurs in finite time and find an upper bound for the blow-up time.

On the problem of superconvergence of finite element method algorithms

The coincidence of an approximate solution to the boundary value problem for an ordinary differential equation with the exact solution at mesh nodes is proved for a certain class of the generalized finite element methods.