Regularity and time behavior of the solutions to weak monotone parabolic equations

In this paper, we study the behavior in time of the solutions for a class of parabolic problems including the p-Laplacian equation and the heat equation. Either the case of singular or degenerate equations is considered. The initial datum $$u_0$$ u 0 is a summable function and a reaction term f is present in the problem. We prove that, despite the lack of regularity of $$u_0$$ u 0 , immediate regularization of the solutions appears for data f sufficiently regular and we derive estimates that for zero data f become the known decay estimates for these kinds of problems. Besides, even if f is not regular, we show that it is possible to describe the behavior in time of a suitable class of solutions. Finally, we establish some uniqueness results for the solutions of these evolution problems.

Stability of a weak solution for a hyperbolic system with distributed parameters on a graph

In the work, the stability conditions for a solution of an evolutionary hyperbolic system with distributed parameters on a graph describing the oscillating process of continuous medium in a spatial network are indicated. The hyperbolic system is considered in the weak formulation: a weak solution of the system is a summable function that satisfies the integral identity which determines the variational formulation for the initial-boundary value problem. The basic idea, that has determined the content of this work, is to present a weak solution in the form of a generalized Fourier series and continue with an analysis of the convergence of this series and the series obtained by its single termwise differentiation. The used approach is based on a priori estimates of a weak solution and the construction (by the Fayedo–Galerkin method with a special basis, the system of eigenfunctions of the elliptic operator of a hyperbolic equation) of a weakly compact family of approximate solutions in the selected state space. The obtained results underlie the analysis of optimal control problems of oscillations of netset-like industrial constructions which have interesting analogies with multi-phase problems of multidimensional hydrodynamics.

On sets of singular rotations for translation invariant differentiation bases formed by intervals

We study the dependence of differential properties of an indefinite integral on a rotation of the coordinate system. Namely, the following problem is studied: For a summable function f, what kind of a set may be the set of rotations θ for which {\int f} is not differentiable with respect to the θ-rotation of a given basis B? For translation invariant bases B formed by two-dimensional intervals, some classes of sets of singular rotations are found. In particular, for such bases with symmetric structure, a characterization of at most countable sets of singular rotations is found.

Further results on the neutrix composition of distributions involving the delta function and the function cosh+-1(x1/r+1)$\cosh _ + ^{ - 1}\left( {{x^{1/r}} + 1} \right)$

The neutrix composition F(f (x)) of a distribution F(x) and a locally summable function f (x) is said to exist and be equal to the distribution h(x) if the neutrix limit of the sequence {Fn(f (x))} is equal to h(x), where Fn(x) = F(x) * δn(x) and {δn(x)} is a certain sequence of infinitely differentiable functions converging to the Dirac delta-function (x). The function $\cosh _ + ^{ - 1}\left( {x + 1} \right)$ is defined by$$\cosh _ + ^{ - 1}\left( {x + 1} \right) = H\left( x \right){\cosh ^{ - 1}}\left( {\left| x \right| + 1} \right),$$where H(x) denotes Heaviside’s function. It is then proved that the neutrix composition ${\delta ^{(s)}}\left[ {\cosh _ + ^{ - 1}\left( {{x^{1/r}} + 1} \right)} \right]$] exists and$${\delta ^{(s)}}\left[ {\cosh _ + ^{ - 1}\left( {{x^{1/r}} + 1} \right)} \right] = \sum\limits_{k = 0}^{s - 1} {\sum\limits_{j = 0}^{kr + r - 1} {\sum\limits_{i = 0}^j {{{{{( - 1)}^{kr + r + s - j - 1}}r} \over {{2^{j + 2}}}}\left( {\matrix{{kr + r - 1} \cr j \cr } } \right)} } } \left( {\matrix{j \cr i \cr } } \right)\left[ {{{\left( {j - 2i + 1} \right)}^s} - {{\left( {i - 2i - 1} \right)}^s}} \right]{\delta ^{(k)}}(x),$$for r, s = 1, 2, . . . . Further results are also proved.Our results improve, extend and generalize the main theorem of [Fisher B., Al-Sirehy F., Some results on the neutrix composition of distributions involving the delta function and the function cosh−1+(x + 1), Appl. Math. Sci. (Ruse), 2014, 8(153), 7629–7640].

On the neutrix composition of x−−slnmx− and ln(1 + x+)

The neutrix composition [Formula: see text], [Formula: see text] is a distribution and [Formula: see text] is a locally summable function, is defined as the neutrix limit of the sequence [Formula: see text], where [Formula: see text] and [Formula: see text] is a certain sequence of infinitely differentiable functions converging to the Dirac delta-function [Formula: see text]. The neutrix composition of the distributions [Formula: see text] and [Formula: see text] is evaluated for [Formula: see text] Further related results are also deduced.

The composition of the distributions x−λlnsx− and x+−r/λ

The composition [Formula: see text] of a distribution [Formula: see text] and a locally summable function [Formula: see text] is defined as the neutrix limit of the regular sequence [Formula: see text] In this paper, we prove that the neutrix composition of the distributions [Formula: see text] and [Formula: see text] exists and equals [Formula: see text] for [Formula: see text], [Formula: see text] [Formula: see text] and [Formula: see text] where [Formula: see text] denotes the Beta function, [Formula: see text] for [Formula: see text] and [Formula: see text] for [Formula: see text] and [Formula: see text] for [Formula: see text]

Space monitoring of the earth on the presence of solid domestic wastes using a discrete orthogonal transforms

The paper investigates multivariate Wavelet Haar?s series. To study on the correctness is made by means of Tikhonov?s method. A theorem on stability and uniform convergence of a regularized summable function of the wavelet-Haar?s series functions in Lipschitz class with approximate coefficients is proved. An experiment confirms the validity of Tikhonov?s method using space monitoring of waste disposal facilities is conducted as an example. Namely, the decoding of space images-images using N-dimensional Haar?s wavelet transform is used.

Some remarks on incomplete gamma type function γ*(α, x_)

The incomplete gamma type function ?*(?, x_) is defined as locally summable function on the real line for ?>0 by ?*(?,x_) = {?x0 |u|?-1 e-u du, x?0; 0, x > 0 = ?-x_0 |u|?-1 e-u du the integral divergining ? ? 0 and by using the recurrence relation ?*(? + 1,x_) = -??*(?,x_) - x?_ e-x the definition of ?*(?, x_) can be extended to the negative non-integer values of ?. Recently the authors [8] defined ?*(-m, x_) for m = 0, 1, 2,... . In this paper we define the derivatives of the incomplete gamma type function ?*(?, x_) as a distribution for all ? < 0.

On the Composition and Neutrix Composition of the Delta Function with the Hyperbolic Tangent and Its Inverse Functions

LetFbe a distribution inD'and letfbe a locally summable function. The compositionF(f(x))ofFandfis said to exist and be equal to the distributionh(x)if the limit of the sequence{Fn(f(x))}is equal toh(x), whereFn(x)=F(x)*δn(x)forn=1,2,…and{δn(x)}is a certain regular sequence converging to the Dirac delta function. It is proved that the neutrix compositionδ(rs-1)((tanhx+)1/r)exists andδ(rs-1)((tanhx+)1/r)=∑k=0s-1∑i=0Kk((-1)kcs-2i-1,k(rs)!/2sk!)δ(k)(x)forr,s=1,2,…, whereKkis the integer part of(s-k-1)/2and the constantscj,kare defined by the expansion(tanh-1x)k={∑i=0∞(x2i+1/(2i+1))}k=∑j=k∞cj,kxj, fork=0,1,2,….Further results are also proved.

On the Neutrix Composition of the Delta and Inverse Hyperbolic Sine Functions

LetFbe a distribution inD'and letfbe a locally summable function. The compositionF(f(x))ofFandfis said to exist and be equal to the distributionh(x)if the limit of the sequence{Fn(f(x))}is equal toh(x), whereFn(x)=F(x)*δn(x)forn=1,2,…and{δn(x)}is a certain regular sequence converging to the Dirac delta function. In the ordinary sense, the compositionδ(s)[(sinh⁡-1x+)r]does not exists. In this study, it is proved that the neutrix compositionδ(s)[(sinh⁡-1x+)r]exists and is given byδ(s)[(sinh⁡-1x+)r]=∑k=0sr+r-1∑i=0k(ki)((-1)krcs,k,i/2k+1k!)δ(k)(x), fors=0,1,2,…andr=1,2,…, wherecs,k,i=(-1)ss![(k-2i+1)rs-1+(k-2i-1)rs+r-1]/(2(rs+r-1)!). Further results are also proved.