On the solvability of a nonlinear pseudoparabolic problem

This paper is concerned with the study of an initial boundary value problem for a nonlinear second order pseudoparabolic equation arising in the unidirectional flow of a thermodynamic compatible third grade fluid. We establish some a priori bounds for the solution and prove its existence.     

The problem of the solvability of nonlinear two-point boundary-value problems

We establish sufficient conditions for the solvability of boundary-value problems of the form $$\begin{gathered} u'' = f(t,u,u'); \hfill \\ \begin{array}{*{20}c} {(u(0),} & {u'(0)) \in S_0 ,} & {(u(1),} & {u'(1)) \in S_1 .} \\ \end{array} \hfill \\ \end{gathered} $$      

Local solvability of a problem of nonlinear elasticity theory

One considers an integral functional depending only on the trace of the metric tensor induced by a mapping of ann-dimensional domain into ⁿ. One seeks a mapping which minimizes this functional. Under a well-defined smallness in the boundary conditions, one proves the existence of an infinite set of critical points of the functional. Under additional restrictions, one discusses an existence theorem and the character of the extremum. The convexity of the functional is not assumed. Such functionals are encountered in elasticity theory.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 110, pp. 163–173, 1981.     

On solvability of a periodic problem for a nonlinear telegraph equation

The time-periodic problem is studied for a nonlinear telegraph equation with the Dirichlet–Poincaré boundary conditions. The questions are considered of existence and smoothness of solutions to this problem.     

The problem of forward motion of a hovercraft: asymptotic analysis of a solution

The linear unsteady problem describing the forward motion of a hovercraft with an oscillating forward velocity is considered. A two-term asymptotic representation of a solution is derived provided that the oscillation period is small. Estimates for the remainder in Sobolev spaces and some hydrodynamical consequences are also obtained. Bibliography: 17 titles. Illustrations: 1 figure. Dedicated to Nina Nikolaevna Uraltseva Translated from Problemy Matematicheskogo Analiza, 40, May 2009, pp. 91–104.     

Local solvability of the cauchy problem of a fifth-order nonlinear dispersive equation

The solvability of the fifth-order nonlinear dispersive equation δtu＋au （δxu）^2＋βδx^3u＋γδx^5u = 0 is studied. By using the approach of Kenig, Ponce and Vega and some Strichartz estimates for the corresponding linear problem,it is proved that if the initial function u0 belongs to H^5（R） and s〉1/4,then the Cauchy problem has a unique solution in C（[-T,T],H^5（R）） for some T〉0.     

On the solvability of a nonlinear integro-differential equation arising in the income distribution problem

A class of the Hammerstein nonlinear integro-differential equations arising in the theory of income distribution is considered. The existence of solutions to these equations in the Sobolev space is proved. An application model described by such an equation is considered, and an algorithm for its solution is proposed. Numerical results are also presented.     

The solvability of a boundary-value periodic problem

In the space of functions B _a³⁺={g(x, t)=−g(−x, t)=g(x+2π, t)=−g(x, t+T₃/2)=g(x, −t)}, we establish that if the condition aT ₃ (2s−1)=4πk, (4πk, a (2s−1))=1, k ∈ ℤ, s ∈ ℕ, is satisfied, then the linear problem u _u −a ² u _xx =g(x, t), u(0, t)=u(π, t)=0, u(x, t+T ₃ )=u(x, t), ℝ², is always consistent. To prove this statement, we construct an exact solution in the form of an integral operator. Ternopol’ Pedagogical Institute, Ternopol’. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 2, pp. 302–308, Feburary, 1997 Ternopol’ Pedagogical Institute, Ternopol’. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 2, pp. 302–308, Feburary, 1997     

The unique solvability of a problem without initial conditions for linear and nonlinear elliptic-parabolic equations

The existence and the uniqueness of generalized solutions of a problem without initial conditions are established for linear and nonlinear anisotropic elliptic-parabolic second-order equations in domains unbounded in spatial variables. We put the restrictions on the behavior of solutions of the problem and the growth of its initial data at infinity. The equations have the nonlinearity exponents depending on points of the domain of definition and the direction of differentiation. Their weak solutions are taken from generalized Lebesgue–Sobolev spaces.     

On the solvability of a boundary value problem for nonlinear wave equations in angular domains

For a one-dimensional wave equation with a weak nonlinearity, we study the Darboux boundary value problem in angular domains, for which we analyze the existence and uniqueness of a global solution and the existence of local solutions as well as the absence of global solutions.     

Nonlocal solvability of the Cauchy problem for second-order nonlinear parabolic equations

We prove the existence of smooth solutions of the Cauchy problem for some second-order nonlinear parabolic equations subject to natural smoothness conditions on the right side of the equation and on the initial function.Translated from Matematicheskie Zametki, Vol. 15, No. 4, pp. 581–586, April, 1974.     

On the solvability of an abstract Cauchy problem

We consider the Cauchy problem for a first-order linear inhomogeneous differential equation for functions ranging in a Banach space in the case of a sectorial operator coefficient. We find conditions for the solvability of the Cauchy problem for the case in which the right-hand side is not necessarily a Hölder function.     

On the solvability of a boundary-value problem for an elliptic equation

Consider a boundary-value problem for a second-order linear elliptic equation in a bounded domain. The coefficient of the required function is nonpositive everywhere in the domain, except for a small neighborhood of an interior point. The following question arises: Under what constraints on this coefficient in the given small domain do the statements on the existence and uniqueness of the solution of the first boundary-value problem remain valid?     

The problem of the motion of a solid in an ideal fluid. Integration of the Clebsch's case

This paper deals with a geometric approach to the integration of Clebsch's case of equations describing the motion of a solid body in an ideal fluid. This problem is defined by a nonlinear system of 6 differential equations admitting 4 polynomial first integrals. We show that the intersection of surface levels of these integrals can be completed to an abelian surface, i.e., a 2-dimensional algebraic torus. Also, we prove that the problem can be linearized, i.e., can be written in terms of abelian integrals, on a Prym variety of a genus 3 curve obtained naturally. Received August 1998     

Unique solvability of the Dirichlet problem for an ultrahyperbolic equation in a ball

We obtain a criterion in terms of zeros of Jacobi polynomials for the uniqueness of the solution of the first boundary value problem for an ultrahyperbolic equation in a ball. The nonuniqueness in the Dirichlet problem proves to occur if and only if the coefficient of the equation belongs to a countable dense subset of the real line.     

On the solvability of a nonlinear boundary value problem arising in the theory of growing cell populations

In this paper we establish some results regarding the existence of solution on L₁ spaces to a nonlinear boundary value problem originally proposed by Lebowitz and Rubinow (J. Math. Biol. 1974; 1 :17–36) to model an age‐structured proliferating cell population. Our approach, based on topological methods, uses essentially the specific properties of weakly compact sets on L₁ spaces. Our results provide positive answers to the questions posed in Jeribi (Nonlinear Anal. Real World Appl. 2002; 3 :85–105). Copyright © 2005 John Wiley & Sons, Ltd.