Volume 3 Issue #3 (September 2011)

  • Andriyanova E.R., Mukminov F.Kh. The lower estimate of decay rate of solutions for doubly nonlinear parabolic equations. Pp. 3 - 14
    Abstract: Existence of strong solution to doubly nonlinear parabolic equation is established on unbounded domains by the method of Galerkin's approximations. In early publications existence was proved usually on bounded domains by approximating the evolution part of the equation by finite differences. Usage of Galerkin's approximations makes it possible to prove the second integral identity. On the basis of the identity, the bottom estimate of decay rate of the solution norm is proved on bounded domains. Similar estimates for quasilinear parabolic equations were established earlier by Tedeev A. F. and Alikakos N., Rostmanian R.
  • Zhiber A.V., Kostrigina O.S. Goursat problem for nonlinear hyperbolic systems with integrals of the first and second order. Pp. 65 - 77
    Abstract: We consider the Goursat problem for one class of nonlinear hyperbolic systems of equations of the form \[u^i_{xy}=F^i(u, u_x, u_y),\qquad i=1,2,\quad u=(u^1,u^2),\] with integrals of the first and second order \[\begin{gather*} \omega^1(u^1,u^2,u^1_x,u^2_x),\ \omega^2(u^1,u^2,u^1_x,u^2_x,u^1_{xx},u^2_{xx}),\quad(\overline D(\omega^1)=\overline D(\omega^2)=0),\\ \overline\omega^1(u^1,u^2,u^1_y,u^2_y),\ \overline\omega^2(u^1,u^2,u^1_y,u^2_y,u^1_{yy},u^2_{yy}),\quad(D(\overline\omega^1)=D(\overline\omega^2)=0). \end{gather*}\] Explicit formulas for the solutions of the Goursat problem with the data set on the characteristics \[\begin{gather*} u^1(x_0,y)=\phi_1(y),\quad u^2(x_0,y)=\phi_2(y),\\ u^1(x,y_0)=\psi_1(x),\quad u^2(x,y_0)=\psi_2(x). \end{gather*}\]
  • Mirzoev K.A., Safonova T.A. The singular Sturm–Liouville operators with nonsmooth potentials in a space of vector functions. Pp. 102 - 115
    Abstract: This paper deals with the Sturm-Liouville operators generated on the semi-axis by the differential expression \(l[y]=-(y'-Py)'-P(y'-Py)-P^2y\), where \(′\) is a derivative in terms of the theory of distributions and \(P\) is a real-valued symmetrical matrix with elements \(p_{ij}\in L^2_{loc}(R_+)\) (\(i,j=1,2,\dots,n\)). The minimal closed symmetric operator \(L_0\) generated by this expression in the Hilbert space \(\mathcal L^2_n(R_+)\) is constructed. Sufficient conditions of minimality and maximality of deficiency numbers of the operator \(L_0\) in terms of elements of a matrix \(P\) are presented. Moreover, it is established, that the condition of maximality of deficiency numbers of the operator \(L_0\) (in the case when elements of the matrix \(P\) are step functions with an infinite number of jumps) is equivalent to the condition of maximality of deficiency numbers of the operator generated by a generalized Jacobi matrix in the space \(l^2_n\).
  • Rakhmatullina Zh.G. The Fatou set of an entire function with the Fejér gaps. Pp. 116 - 121
    Abstract: The paper considers the Fatou set of an entire transcendental function, i.e. the largest open set of the complex plane where the family of iterations of the given function forms a normal family. We assume that the entire function, in general, is of an infinite order. We give the sufficient condition on the indexes of the series (it is stronger than the Fejér gap condition), under which every component of the Fatou set is bounded. The same result under stronger restrictions was earlier obtained by Yu. Wang.
  • Smailov E.S., Takuadina A.I. About the unimprobality of the limiting embedding theorem for different metrics in the Lorentz spaces with Hermite's weight. Pp. 135 - 145
    Abstract: In this article we obtained inequality of different metrics in the Lorentz spaces with Hermit's weight for multiple algebraic polynomials. On this basis we established a sufficient condition of embedding of different metrics in the Lorenz spaces with Hermite's weight. Its unimprobality is shown in terms of the “extreme function”. \(f\in L_{p,\theta}(\mathbb R_n;\rho_n)\), \(1\leq p<+\infty\), \(1\leq\theta\leq+\infty\). The sequense \(t\{l_k\}_{k=0}^{+\infty}\subset\mathbb N\) is such that \(l_0=1\) and \(l_{k+1}\cdot l_k^{-1}>a_0>1\), \(\forall k\in\mathbb Z^+\). \(f(\bar x)=\sum_{k=0}^{+\infty}\Delta_{l_k,\dots,l_k}(f;\bar x)\) is some presentation of the functions in the metric \(L_{p,\theta}(\mathbb R_n;\rho_n)\), where \[\Delta_{l_0,\dots,l_0}(f;\bar x)=T_{1,\dots,1},\Delta_{l_k,\dots,l_k}(f;\bar x)=T_{l_k,\dots,l_k}(\bar x)-T_{l_{k-1},\dots,l_{k-1}}(\bar x)\], \(\forall k\in\mathbb N\). Here \[T_{l_k,\dots,l_k}(\bar x)=\sum_{m_1=0}^{l_k-1}\dots\sum_{m_n=0}^{l_k-1}a_{m_1,\dots,m_n}\prod^n_{i=1}x^{m_i}_i\] are algebraic polynomials for all \(k\in\mathbb Z^+\). \(1^0\). If the series \[A(f)_{p\theta}=\sum_{k=0}^{+\infty}l_k^{\tau(\frac n{2p}-\frac n{2q})}\|\Delta_{l_k,\dots,l_k}(f)\|_{L_{p,\theta}(\mathbb R_n;\rho_n)}^\tau\] converge under some \(q\) and \(\tau\): \(p0\): \(p<(q-\varepsilon)