Solvability and Stability of the Inverse Problem for the Quadratic Differential Pencil

The inverse spectral problem for the second-order differential pencil with quadratic dependence on the spectral parameter is studied. We obtain sufficient conditions for the global solvability of the inverse problem, prove its local solvability and stability. The problem is considered in the general case of complex-valued pencil coefficients and arbitrary eigenvalue multiplicities. Studying local solvability and stability, we take the possible splitting of multiple eigenvalues under a small perturbation of the spectrum into account. Our approach is constructive. It is based on the reduction of the non-linear inverse problem to a linear equation in the Banach space of infinite sequences. The theoretical results are illustrated by numerical examples.

On the local well-posedness of the nonlinear heat equation associated to the fractional Hermite operator in modulation spaces

In this note we consider the nonlinear heat equation associated to the fractional Hermite operator $$H^\beta =(-\Delta +|x|^2)^\beta $$ H β = ( - Δ + | x | 2 ) β , $$0<\beta \le 1$$ 0 < β ≤ 1 . We show the local solvability of the related Cauchy problem in the framework of modulation spaces. The result is obtained by combining tools from microlocal and time-frequency analysis. As a byproduct, we compute the Gabor matrix of pseudodifferential operators with symbols in the Hörmander class $$S^m_{0,0}$$ S 0 , 0 m , $$m\in \mathbb {R}$$ m ∈ R .

A new approach to the inverse discrete transmission eigenvalue problem

<p style='text-indent:20px;'>A discrete analog is considered for the inverse transmission eigenvalue problem, having applications in acoustics. We provide a well-posed inverse problem statement, develop a constructive procedure for solving this problem, prove uniqueness of solution, global solvability, local solvability, and stability. Our approach is based on the reduction of the discrete transmission eigenvalue problem to a linear system with polynomials of the spectral parameter in the boundary condition.</p>

On the Cauchy problem for implicit differential equations of higher orders

The article is devoted to the study of implicit differential equations based on statements about covering mappings of products of metric spaces. First, we consider the system of equations Φ_i (x_i,x_1,x_2,…,x_n )=y_i, i=(1,n,) ̅ where 〖 Φ〗_i: X_i×X_1×… ×X_n→Y_i, y_i∈Y_i, X_i and Y_i are metric spaces, i=(1,n) ̅. It is assumed that the mapping 〖 Φ〗_i is covering in the first argument and Lipschitz in each of the other arguments starting from the second one. Conditions for the solvability of this system and estimates for the distance from an arbitrary given element x_0∈X to the set of solutions are obtained. Next, we obtain an assertion about the action of the Nemytskii operator in spaces of summable functions and establish the relationship between the covering properties of the Nemytskii operator and the covering of the function that generates it. The listed results are applied to the study of a system of implicit differential equations, for which a statement about the local solvability of the Cauchy problem with constraints on the derivative of a solution is proved. Such problems arise, in particular, in models of controlled systems. In the final part of the article, a differential equation of the n-th order not resolved with respect to the highest derivative is studied by similar methods. Conditions for the existence of a solution to the Cauchy problem are obtained.

An Initial Problem for a Class of Weakly Degenerate Semilinear Equations with Lower Order Fractional Derivatives

An initial value problem is studied for a class of evolutionary equations with a weak degeneration, which are nonlinear with respect to lower order fractional Gerasimov – Caputo derivatives. The linear part of the equations contains a respectively bounded pair of operators. Unique local solvability is proved in the case of a nonlinear operator depending on elements of the degeneration space only. Examples of an equation and a system of partial differential equations are given, the initial-boundary value problems for which are reduced to the initial problem for an equation in a Banach space of the studied class.