Blow-up for one Sobolev problem: Theoretical approach and numerical analysis

Lukyanenko

et al. 2018

Self Cite

The initial‐boundary value problem for an equation of ion sound waves in plasma is considered. A theorem on nonextendable solution is proved. The blow‐up phenomena are studied. The sufficient blow‐up conditions and the blow‐up time are analysed by the method of the test functions. This analytical a priori information is used in the numerical experiments, which are able to determine the process of the solution's blow‐up more accurately.

\{\begin{matrix} u false(0, t false) = 0, u_{x} false(0, t false) = 0 t \in false[0, + \infty false), \\ u false(x, 0 false) = u_{0} false(x false), \frac{\partial}{\partial t} u false(x, 0 false) = u_{1} false(x false), x \in false[0, l false] . \end{matrix}

We prove a theorem on the nonextendable solution.…”

Section: Introductionmentioning

confidence: 60%

On the blow‐up phenomena for a 1‐dimensional equation of ion sound waves in a plasma: Analytical and numerical investigation

Lukyanenko

et al. 2018

Self Cite

“…It is clear that (9) is similar to the model equation (1) with parameters " D r 2 d and D 1 . The linearized model (for the large Debye screening radius " 1) was considered in [5,6]; here, we consider the general nonlinear problem.…”

Section: Introductionmentioning

confidence: 62%

“…The linearized model (for the large Debye screening radius " 1) was considered in [5,6]; here, we consider the general nonlinear problem. The test function method that is used in the analytical part (Section 2) is often applied for power and exponential nonlinearities and various boundary conditions (see, e.g., [8][9][10]). The method that has been used for the local solvability investigation (Section 2) can be used after model transformation into the integral form and is based on the well-known results of contracting mapping [11].…”

Section: Introductionmentioning

confidence: 99%

Blow‐up phenomena in the model of a space charge stratification in semiconductors: analytical and numerical analysis

Lukyanenko

et al. 2016

The initial‐boundary value problems for a Sobolev equation with exponential nonlinearities, classical, and nonclassical boundary conditions are considered. For this model, which describes processes in crystalline semiconductors, the blow‐up phenomena are studied. The sufficient blow‐up conditions and the blow‐up time are analyzed by the method of the test functions. This analytical a priori information is used in the numerical experiments, which are able to determine the process of the solution's blow‐up more accurately. The model derivation and some questions of local solvability and uniqueness are also discussed. Copyright © 2016 John Wiley & Sons, Ltd.

“…Remark As a possible further evolvement of the present work, we can suggest the development of numerical method (see our papers()) that allows to detect the instantaneous blow‐up. But we have to notice that, in case of absence of a priori information obtained (as in the present paper) analytically, numerical calculation is not a rigorous proof that instantaneous blow‐up really occurs.…”

Section: Cauchy Problemmentioning

confidence: 81%

Instantaneous blow‐up versus local solvability of solutions to the Cauchy problem for the equation of a semiconductor in a magnetic field

Овчинников

2018