Cauchy–Goursat Problem for One-Dimensional Semilinear Wave Equations

We consider the initial-characteristic problem for nonlinear wave equations with positive power nonlinearity source term. Depending on the power of nonlinearity, we investigate the problem on a global existence and blow-up of solutions of initial-characteristic problem. The question on local solvability of the problem is also considered.     

The first Darboux problem for wave equations with a nonlinear positive source term

We consider the first Darboux problem for nonlinear wave equations with positive power nonlinearity source term. Depending on the power of nonlinearity we investigate the problem on a global existence and blow-up of solutions of the first Darboux problem. The question of local solvability of the problem is also considered.     

Global existence of a restoring force realizing a prescribed half-period

We consider an inverse problem to determine a nonlinearity of an autonomous differential equation so that each solution of the equation has a half-period that is a prescribed function of a half-amplitude of the solution. A global existence of the nonlinearity is proved under the assumption that the prescribed function is Lipschitz continuous. A reconstructive way of the nonlinearity is also given.     

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.     

A Small-Parameter Problem from the Theory of Boundary-Value Problems with a Small Nonlinearity

We consider a system of nonlinear equations with a small (in a certain sense) nonlinearity. The existence of a solution is investigated in the presence of a perturbation parameter of small absolute value, and the perturbed solution is required to approach the original solution of the limiting linear system. The sufficient conditions are presented in geometrical and analytical forms. The problem considered in this article arises in connection with nonlinear boundary-value problems for systems of differential equations with a small (in a certain sense) nonlinearity.     

On the solvability of the Cauchy characteristic problem for a nonlinear equation with iterated wave operator in the principal part

We consider one multidimensional version of the Cauchy characteristic problem in the light cone of the future for a hyperbolic equation with power nonlinearity with iterated wave operator in the principal part. Depending on the exponent of nonlinearity and spatial dimension of equation, we investigate the problem on the nonexistence of global solutions of the Cauchy characteristic problem. The question on the local solvability of that problem is also considered.     

Solution blowup for the heat equation with double nonlinearity

We consider a model initial boundary value problem for the heat equation with double nonlinearity. We use a modified Levin method to prove the solution blowup.     

Asymptotics of The Spectrum of a Two-Dimensional Hartree-Type Operator with a Coulomb Self-Action Potential Near the Lower Boundaries of Spectral Clusters

Theoretical and Mathematical Physics - We consider the eigenvalue problem for a perturbed two-dimensional oscillator where the perturbation is an integral Hartree-type nonlinearity with a Coulomb...     

Propagation of Gaussian wave packets in thin periodic quantum waveguides with a nonlocal nonlinearity

We consider the nonlinear Schrödinger equation with an integral Hartree-type nonlinearity in a thin quantum waveguide and study the propagation of Gaussian wave packets localized in the spatial variables. In the case of periodically varying waveguide walls, we establish the relation between the behavior of wave packets and the spectral properties of the auxiliary periodic problem for the one-dimensional Schrödinger equation. We show that for a positive value of the nonlinearity parameter, the integral nonlinearity prevents the packet from spreading as it propagates. In addition, we find situations such that the packet is strongly focused periodically in time and space.     

On existence and nonexistence of global solutions of Cauchy–Goursat problem for nonlinear wave equations

We consider the Cauchy–Goursat initial characteristic problem for nonlinear wave equations with power nonlinearity. Depending on the power of nonlinearity and the parameter in an equation we investigate the problem on existence and nonexistence of global solutions of the Cauchy–Goursat problem. The question on local solvability of the problem is also considered.     

Stability of Solitary Waves for a Generalized Derivative Nonlinear Schrödinger Equation

We consider a derivative nonlinear Schrödinger equation with a general nonlinearity. This equation has a two-parameter family of solitary wave solutions. We prove orbital stability/instability results that depend on the strength of the nonlinearity and, in some instances, on the velocity. We illustrate these results with numerical simulations.     

On a nonlinear inequality on the time scale

In the present paper, we consider a dynamic nonlinear integral inequality with a power-law nonlinearity. We obtain a solution of this inequality for arbitrary nonlinearity exponents exceeding unity. These results can be constructively used in the analysis of stability properties (including nonclassical stability properties) of quasilinear dynamic equations.     

First Darboux problem for nonlinear hyperbolic equations of second order

We study the first Darboux problem for hyperbolic equations of second order with power nonlinearity. We consider the question of the existence and nonexistence of global solutions to this problem depending on the sign of the parameter before the nonlinear term and the degree of its nonlinearity. We also discuss the question of local solvability of the problem.     

On a nonlinear equation with fractional derivative

We consider a nonlinear equation with fractional derivative in which the nonlinearity has the form of a linear combination of convective and nonconvective types. We prove the time-global existence of solutions of the Cauchy problem and find their asymptotics at large times uniformly with respect to the space variable. The proof method is based on a detailed investigation of the behavior of the Green function, which permits one to drop the restriction of smallness of the initial data in the case of a nonlinearity of special form.     

Null Controllability for the Dissipative Semilinear Heat Equation

We consider the exact null controllability problem for the semi- linear heat equation with dissipative nonlinearity in a bounded domain of Rⁿ . The main result of the article asserts that if the nonlinearity is even mildly superlinear, then global null controllability in an arbitrarily short time fails; instead we provide sharp estimates for the controllability time in terms of the size of the initial data.     

Sufficient tests for the existence and uniqueness of a solution of the cauchy problem for a differential equation with a hysteresis nonlinearity

For a differential equation with a hysteresis nonlinearity of general type, which can be time-varying, we obtain sufficient conditions for the existence and uniqueness of a solution of the Cauchy problem similar to the well-known Cauchy-Picard, Peano, Perron, and Rosenblatt theorems for ordinary differential equations. We consider examples in which a test for the solution uniqueness similar to the Perron-Rosenblatt theorem is applied to specific differential equations with hysteresis nonlinearities of the form of the Prandtl model of a viscoelastic fiber and a time-varying nonlinearity.     

The necessary optimality conditions for a nonlinear stationary system whose state functional is not differentiable with respect to the control

We consider a control system described by a nonlinear elliptic equation. Its control-state mapping is extendedly differentiable but not Gateaux differentiable for large values of the domain dimension and the nonlinearity index. We obtain the necessary optimality condition for various state functionals.     

Semiclassical asymptotic spectrum of the two-dimensional Hartree operator near a local maximum of the eigenvalues in a spectral cluster

Theoretical and Mathematical Physics - We consider the eigenvalue problem for the two-dimensional Hartree operator with a small nonlinearity coefficient. We find the asymptotic eigenvalues and...     

Optimization for nonlinear hyperbolic equations without the uniqueness theorem for a solution of the boundary-value problem

We consider the optimal control problem for a system governed by a nonlinear hyperbolic equation without any constraints on the parameter of nonlinearity. No uniqueness theorem is established for a solution to this problem. The control-state mapping of this system is not Gateaux differentiable. We study an approximate solution of the optimal control problem by means of the penalty method.