On a problem of integral geometry related to the Dirichlet problem for an ultrahyperbolic equation

We study the nontrivial solvability of the homogeneous problem of integral geometry on the family of spheres formed by sections of the (n − 1)-dimensional unit sphere by hyperplanes perpendicular to the generators of a given cone. By using a relationship between this problem and the Dirichlet problem for an ultrahyperbolic equation in a ball and a criterion for the failure of uniqueness of the solution of the latter problem in terms of zeros of classical Jacobi polynomials, we obtain a criterion for the uniqueness of the solution of the former problem.     

On the Cauchy problem for a reaction-diffusion equation with a singular nonlinearity

We consider the following Cauchy problem with a singular nonlinearity

(P)

     

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.     

The singular Cauchy problem for a non-linear hyperbolic equation

Summary The author demonstrates the existence of a smooth solution to a singular initial value problem for a quasiliuear hyperbolic equation in two independent variables. The problem is transformed into an equivalent system of integral equations for which a solution is obtained by invoking Schauder’s fixed point theorem. Entrata in Redazione il 19 ottobre 1970.     

Mixed nonlocal problem for a nonlinear singular hyperbolic equation

In this paper, we study a nonlocal mixed problem for a nonlinear hyperbolic equation. Based on some a priori estimates and some density arguments, we prove the well posedness of the associated linear problem. The existence and uniqueness of the weak solution of the nonlinear problem are then established by applying an iterative process based on the obtained results for the linear problem. Copyright © 2009 John Wiley & Sons, Ltd.     

The Robin problem for singular p-Laplacian equation in a cone

We study the behaviour near the boundary angular or conical point of weak solutions to the Robin problem for elliptic quasi-linear second-order equation with the p-Laplacian.     

On solvability of the Cauchy problem for one quasilinear singular functional-differential equation

We consider the Cauchy problem with zero initial conditions for quasilinear singular functional-differential equation of the second order with a delay at singular summand. We obtain sufficient conditions of solvability of the problem.     

On a singular minimizing problem

We investigate the non-homogeneous modular Dirichlet problem Δ _p (·)u(x) = f (x) (where Δ _p (·)u(x) = div(|?u|^p(x-2)?u(x)) from the functional analytic point of view and we prove the stability of the solutions \({\left( {{u_{{p_i}}}} \right)_i}\) of the equation \({\Delta _{{p_i}\left( \cdot \right)}}{u_{{p_i}\left( \cdot \right)}} = f\) as p _i (·) → q(·) via Gamma-convergence of sequence of appropriate functionals.     

ASYMPTOTIC SOLUTION OF SINGULARLY PERTURBED PROBLEM FOR A NONLINEAR SINGULAR EQUATION

The singularly perturbed initial value problem for a nonlinear singular equation is considered. By using a simple and special method the asymptotic behavior of solution is studied.     

Qualitative investigation of a singular Cauchy problem for a functional differential equation

We consider the singular Cauchy problem

Cauchy problem for a second-order hyperbolic operator-differential equation with a singular coefficient

In a Hilbert space, we study the well-posedness of the Cauchy problem for a second-order operator-differential equation with a singular coefficient.