Maxim Olegovich Korpusov

Blow-up in nonlinear sobolev type equations

The monograph is devoted to the study of initial-boundary-value problems for multi-dimensional Sobolev-type equations over bounded domains. The authors consider both specific initial-boundary-value problems and abstract Cauchy problems for first-order (in the time variable) differential equations with nonlinear operator coefficients with respect to spatial variables. The main aim of the monograph is to obtain sufficient conditions for global (in time) solvability, to obtain sufficient conditions for blow-up of solutions at finite time, and to derive upper and lower estimates for the blow-up time. The monograph contains a vast list of references (440 items) and gives an overall view of the contemporary state-of-the-art of the mathematical modeling of various important problems arising in physics. Since the list of references contains many papers which have been published previously only in Russian research journals, it may also serve as a guide to the Russian literature.

Global unsolvability of one-dimensional problems for Burgers-type equations

In this paper, we study the global solvability of well-known equations used to describe nonlinear processes with dissipation, namely, the Burgers equation, the Korteweg–de Vries–Burgers equation, and the modified Korteweg–de Vries–Burgers equation. Using a method due to Pokhozhaev, we obtain necessary conditions for the blow-up of global solutions and estimates of the blow-up time and blow-up rate in bounded and unbounded domains. We also study the effect of linear and nonlinear viscosity on the occurrence of a gradient catastrophe in finite time.

On the blow-up of the solution of an equation related to the Hamilton-Jacobi equation

A new model three-dimensional third-order equation of Hamilton-Jacobi type is derived. For this equation, the initial boundary-value problem in a bounded domain with smooth boundary is studied and local solvability in the strong generalized sense is proved; in addition, sufficient conditions for the blow-up in finite time and sufficient conditions for global (in time) solvability are obtained.

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.

The finite-time blowup of the solution of an initial boundary-value problem for the nonlinear equation of ion sound waves

We obtain blowup conditions for the solutions of initial boundary-value problems for the nonlinear equation of ion sound waves in a hydrogen plasma in the approximation of “hot” electrons and “heavy” ions. A specific characteristic of this nonlinear equation is the noncoercive nonlinearity of the form ∂t|∇u|2, which complicates its study by any energy method. We solve this problem by the Mitidieri–Pohozaev method of nonlinear capacity.

Solution Blowup for Nonlinear Equations of the Khokhlov–Zabolotskaya Type

We consider several nonlinear evolution equations sharing a nonlinearity of the form ∂2u2/∂t2. Such a nonlinearity is present in the Khokhlov–Zabolotskaya equation, in other equations in the theory of nonlinear waves in a fluid, and also in equations in the theory of electromagnetic waves and ion–sound waves in a plasma. We consider sufficient conditions for a blowup regime to arise and find initial functions for which a solution understood in the classical sense is totally absent, even locally in time, i.e., we study the problem of an instantaneous blowup of classical solutions.

Global unsolvability of a nonlinear conductor model in the quasistationary approximation

We study initial-boundary value problems for a model differential equation in a bounded region with a quadratic nonlinearity of a special type typical for the theory of conductors. Using the test function method, we show that such a nonlinearity can lead to global unsolvability with respect to time, which from the physical standpoint means an electrical breakdown of the conductor in a finite time. For the simplest test functions, we obtain sufficient conditions for the unsolvability of the model problems and estimates of the blowup rate and time. With concrete examples, we demonstrate the possibility of using the method for one-, two- and three-dimensional problems with classical and nonclassical boundary conditions. We separately consider the Neumann and Navier problems in bounded RN regions (N ≥ 2).

Градиентная катастрофа в обобщенных уравнениях Бюргерса и Буссинеска@@@Gradient blow-up in the generalized Burgers and Boussinesq equations

В работе исследуется влияние градиентной нелинейности на глобальную разрешимость начально-краевых задач для обобщенного уравнения Бюргерса и улучшенного уравнения Буссинеска, используемых для описания одномерных волновых процессов в средах с диссипацией и дисперсией. Для широкого класса начальных данных получены достаточные условия глобальной неразрешимости и оценка на времена разрушений. На примере уравнения Буссинеска предложена модификация метода нелинейной емкости удобная с практической точки зрения и позволяющая оценить скорость разрушения. С помощью метода сжимающих отображений рассмотрены вопросы о возможности мгновенного разрушения и разрешимости на малых временах. Библиография: 15 наименований.