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

Abstract

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.

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

“…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.…”

“…Now, we consider the critical case q = 3; now, c 1 , c 2 , and c 3 do not tend to 0 as R → + ∞ but remain bounded. Therefore, the estimate (28) and the limit relation (29) obtained by using Beppo Levi's theorem remain valid, but we cannot pass to the limits in the inequality (24) and get (30). We modify our reasonings.…”

“…Applying it to the first two terms in the left-hand side of the inequality (24), we obtain the following inequalities:…”

“…Thus, the theorem is completely proved. [20][21][22][23][24] ) 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.…”

On Some Features of the Numerical Solving of Coefficient Inverse Problems for an Equation of the Reaction-Diffusion-Advection-Type with Data on the Position of a Reaction Front

“…Thus, it becomes necessary to diagnose the existence of a solution on the entire domain Ũ. To do this, we used the method of the numerical diagnostics of the existence of a solution based on the method of refining grids [55,56,[59][60][61].…”