2019
Том 71
№ 11

# Feller M. N.

Articles: 14
Article (Russian)

### Boundary-value problems for a nonlinear hyperbolic equation with Levy Laplaciana

Ukr. Mat. Zh. - 2012. - 64, № 11. - pp. 1492-1499

We present solutions of the boundary-value problem $U(0, x) = u_0, \;U(t, 0) = u_1$, and the external boundary-value problem $U(0, x) = v_0,\; U(t, x)|_{Γ} = v_1,\; \lim_{||x||_H→∞} U(t, x) = v_2$ for the nonlinear hyperbolic equation $$\frac{∂^2U(t, x)}{∂t^2} + α(U(t, x)) \left[\frac{∂U(t, x)}{∂t}\right]^2 = ∆_LU(t, x)$$ with infinite-dimensional Levy Laplacian $∆_L$.

Article (Russian)

### Boundary-value problems for a nonlinear hyperbolic equation with divergent part and Levy Laplacian

Ukr. Mat. Zh. - 2012. - 64, № 2. - pp. 237-244

We propose an algorithm for the solution of the boundary-value problem $U(0,x) = u_0,\;\; U(t, 0) = u_1$ and the external boundary-value problem $U(0, x) = v_0, \;\;U(t, x) |_{\Gamma} = v_1, \;\; \lim_{||x||_H \rightarrow \infty} U(t, x) = v_2$ for the nonlinear hyperbolic equation $$\frac{\partial}{\partial t}\left[k(U(t,x))\frac{\partial U(t,x)}{\partial t}\right] = \Delta_L U(t,x)$$ with divergent part and infinite-dimensional Levy Laplacian $\Delta_L$.

Article (Russian)

### Boundary-value problems for a nonlinear parabolic equation with Lévy Laplacian resolved with respect to the derivative

Ukr. Mat. Zh. - 2010. - 62, № 10. - pp. 1400–1407

We present the solutions of boundary-value and initial boundary-value problems for a nonlinear parabolic equation with Lévy Laplacian $∆_L$ resolved with respect to the derivative $$\frac{∂U(t,x)}{∂t}=f(U(t,x),Δ_LU(t,x))$$ in fundamental domains of a Hilbert space.

Article (Russian)

### Boundary-value problems for the wave equation with Lévy Laplacian in the Gâteaux class

Ukr. Mat. Zh. - 2009. - 61, № 11. - pp. 1564-1574

We present the solutions of the initial-value problem in the entire space and the solutions of the boundary-value and initial-boundary-value problems for the wave equation $$\frac{∂^2U(t,x)}{∂x^2} = Δ_LU(t,x)$$ with infinite-dimensional Lévy Laplacian $Δ_L$ in the class of Gâteaux functions.

Article (Russian)

### Notes on infinite-dimensional nonlinear parabolic equations

Ukr. Mat. Zh. - 2000. - 52, № 5. - pp. 690-701

We present a method for the solution of the Cauchy problem for three broad classes of nonlinear parabolic equations $$\frac{{\partial U\left( {t,x} \right)}}{{\partial t}} = f\left( {\Delta _L U\left( {t,x} \right)} \right), \frac{{\partial U\left( {t,x} \right)}}{{\partial t}} f\left( {t,\Delta _L U\left( {t,x} \right)} \right),$$ and $$\frac{{\partial U\left( {t,x} \right)}}{{\partial t}} = f\left( {U\left( {t,x} \right), \Delta _L U\left( {t,x} \right)} \right)$$ with the infinite-dimensional Laplacian ΔL.

Brief Communications (Russian)

### The riquier problem for a nonlinear equation unresolved with respect to the Lévy iterated laplacian

Ukr. Mat. Zh. - 1999. - 51, № 3. - pp. 423–427

We present a method of solving for the nonlinear equationf(U(x),Δ L 2 U(x)) = Δ L U(x) (Δ L is an infinite-dimensional Laplacian) unresolved with respect to an iterated infinite-dimensional Laplacian and for the Riquier problem for this equation.

Brief Communications (Russian)

### Riquier problem for a nonlinear equation resolved with respect to the iterated Levi Laplacian

Ukr. Mat. Zh. - 1998. - 50, № 11. - pp. 1574–1577

Solutions are found for the nonlinear equation Δ L 2 U(x) = f(U(x)) (here, Δ L is an infinite-dimensional Laplacian) which is solved with respect to the iterated infinite-dimensional Laplacian. The Riquier problems are stated for an equation of this sort.

Brief Communications (Russian)

### On a nonlinear equation unsolved with respect to the levy laplacian

Ukr. Mat. Zh. - 1996. - 48, № 5. - pp. 719-721

We propose a method for the solution of the nonlinear equationf(U(x),ΔU(x))=F(x) (Δ L is an infinite-dimensional Laplacian, Δ L U(x)=γ, γ≠0) unsolved with respect to the infinite-dimensional Laplacian, and for the solution of the Dirichlet problem for this equation.

Brief Communications (Russian)

### New condition of harmonicity of functions of infinitely many variables (translation nonpositive case)

Ukr. Mat. Zh. - 1994. - 46, № 11. - pp. 1602-1605

A criterion of harmonicity of functions in a Hilbert space is given in the case of nonnegative second derivatives without using an assumption that they are mutually independent. This assumption is replaced by a weaker condition.

Brief Communications (Russian)

### Necessary and sufficient conditions of harmonicity of functions of infinitely many variables (Jacobian case)

Ukr. Mat. Zh. - 1994. - 46, № 6. - pp. 785–788

A criterion of harmonicity of functions in a Hilbert space is given in the case of weakened mutual dependence of the second derivatives.

Article (Ukrainian)

### Supply of harmonic functions of an infinite number of variables. II

Ukr. Mat. Zh. - 1990. - 42, № 12. - pp. 1687–1693

Article (Ukrainian)

### Self-adjointness of a nonsymmetrized infinite-dimensional Laplace-Levy operator

Ukr. Mat. Zh. - 1989. - 41, № 7. - pp. 997-1001

Article (Ukrainian)