Nonlinear Equations of the Theory of Ion-Sound Plasma Waves

New high-order nonlinear equations of the Sobolev type describing ion-sound plasma waves in an external electric or magnetic field are derived. Despite the cumbersome form of the equations, research techniques are developed for corresponding initial and initial-boundary value problems. Specifically, based on the present results, sufficient conditions for blow-up formation will be proposed later

Blow-up and global solubility in the classical sense of the Cauchy problem for a formally hyperbolic equation with a non-coercive source

We consider an abstract Cauchy problem with non-linear operator coefficients and prove the existence of a unique non-extendable classical solution. Under certain sufficient close-to-necessary conditions, we obtain finite-time blow-up conditions and upper and lower bounds for the blow-up time. Moreover, under certain sufficient close-to-necessary conditions, we obtain a result on the existence of a global-in-time solution independently of the size of the initial functions. © 2020 Russian Academy of Sciences (DoM) and London Mathematical Society

Blow-up of Solutions of Nonclassical Nonlocal Nonlinear Model Equations

Abstract: For a nonlinear nonlocal operator differential equation of the first order, an abstract Cauchy problem is considered that is a generalization of certain model physical examples. For this problem, the existence of a nonextendable (in time) classical solution is proved. Additionally, finite-time blow-up results are obtained under certain sufficient conditions, and bilateral estimates for the blow-up time are derived. Finally, under certain conditions, the problem is proved to be globally well posed regardless of the value of the initial function. © 2019, Pleiades Publishing, Ltd

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. © 2017, Pleiades Publishing, Ltd

Blow-up instability in non-linear wave models with distributed parameters

We consider two model non-linear equations describing electric oscillations in systems with distributed parameters on the basis of diodes with non-linear characteristics. We obtain equivalent integral equations for classical solutions of the Cauchy problem and the first and second initial-boundary value problems for the original equations in the half-space. Using the contraction mapping principle, we prove the local-in-time solubility of these problems. For one of these equations, we use the Pokhozhaev method of non-linear capacity to deduce a priori bounds giving rise to finite-time blow-up results and obtain upper bounds for the blow-up time. For the other, we use a modification of Levine's method to obtain sufficient conditions for blow-up in the case of sufficiently large initial data and give a lower bound for the order of growth of a functional with the meaning of energy. We also obtain an upper bound for the blow-up time. © 2020 RAS(DoM) and LMS

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. © 2017, Pleiades Publishing, Ltd

Мгновенное разрушение versus локальная разрешимость задачи Коши для двумерного уравнения полупроводника с тепловым разогревом

We consider the Cauchy problem for a model third-order partial differential equation with non-linearity of the form\r\n$|\nabla u|^q$. We prove that for $q\in(1,2]$ the Cauchy problem in $\mathbb{R}^2$ has no local-in-time weak\r\nsolution for a large class of initial functions, while for $q>2$ a local weak solution exists.Рассматривается задача Коши для одного модельного уравнения третьего порядка\r\nв частных производных с нелинейностью вида $|\nabla u|^q$. В работе доказано, что при $q\in(1,2]$ локального во времени слабого решения задачи Коши в $\mathbb{R}^2$ нет для достаточно широкого класса начальных функций, в то время как при $q>2$ локальное слабое решение существует.Библиография: 20 наименований

Potential Theory and Schauder Estimate in Hölder Spaces for (3 + 1)-Dimensional Benjamin–Bona–Mahoney–Burgers Equation

Abstract: The Cauchy problem for the well-known Benjamin–Bona–Mahoney–Burgers equation in the class of Hölder initial functions from C2+α(R3) α (0,1] is considered. For such initial functions, it is proved that the Cauchy problem has a unique time-unextendable classical solution in the classC(1)([0,T];C2+λ(R3)) for any T (0T0) moreover, either T0 = +∞ T0 < +∞ and, in the latter case, T(0) is the solution blow-up time. To prove the solvability of the Cauchy problem, we examine volume and surface potentials associated with the Cauchy problem in Hölder spaces. Finally, a Schauder estimate is obtained. © 2021, Pleiades Publishing, Ltd

Potential Theory for a Nonlinear Equation of the Benjamin–Bona–Mahoney–Burgers Type

Abstract: For the linear part of a nonlinear equation related to the well-known Benjamin–Bona–Mahoney–Burgers (BBMB) equation, a fundamental solution is constructed, which is combined with the second Green formula to obtain a third Green formula in a bounded domain. Then a third Green formula in the entire space is derived by passage to the limit in some class of functions. The properties of the potentials entering the Green formula in the entire space are examined. The Cauchy problem for a nonlinear BBMB-type equation is considered. It is proved that finding its classical solution is equivalent to solving a nonlinear integral equation derived from the third Green formula. The unique local-in-time solvability of this integral equation is proved by applying the contraction mapping principle. Next, the local-in-time classical solvability of the Cauchy problem is proved using the properties of potentials. Finally, the nonlinear capacity method is used to obtain a global-in-time a priori estimate for classical solutions of the Cauchy problem. © 2019, Pleiades Publishing, Ltd