The Cauchy problem for inhomogeneous parabolic Shilov equations

In this paper, we consider the Cauchy problem for parabolic Shilov equations with continuous bounded coefficients. In these equations, the inhomogeneities are continuous exponentially decreasing functions, which have a certain degree of smoothness by the spatial variable. The properties of the fundamental solution of this problem are described without using the kind of equation. The corresponding volume potential, which is a partial solution of the original equation, is investigated. For this Cauchy problem the correct solvability in the class of generalized initial data, which are the Gelfand and Shilov distributions, is determined.

REPRESENTATION OF SOLUTIONS OF KOLMOGOROV TYPE EQUATIONS WITH INCREASING COEFFICIENTS AND DEGENERATIONS ON THE INITIAL HYPERPLANE

The nonhomogeneous model Kolmogorov type ultraparabolic equation with infinitely increasing coefficients at the lowest derivatives as |x| → ∞ and degenerations for t = 0 is considered in the paper. Theorems on the integral representation of solutions of the equation are proved. The representation is written with the use of Poisson integral and the volume potential generated by the fundamental solution of the Cauchy problem. The considered solutions, as functions of x, could infinitely increase as |x| → ∞, and could behave in a certain way as t → 0, depending on the type of the degeneration of the equation at t = 0. Note that in the case of very strong degeneration, the solutions, as functions of x, are bounded. These results could be used to establish the correct solvability of the considered equation with the classical initial condition in the case of weak degeneration of the equation at t = 0, weight initial condition or without the initial condition if the degeneration is strong.

What Drives Household Deforestation Decisions? Insights from the Ecuadorian Lowland Rainforests

Tropical forests, and more concretely, the Amazon Basin and the Chocó-Darién, are highly affected by deforestation activities. Households are the main land-use decision-makers and are key agents for forest conservation and deforestation. Understanding the determinants of deforestation at the household level is critical for conservation policies and sustainable development. We explore the drivers of household deforestation decisions, focusing on the quality of the forest resources (timber volume potential) and the institutional environment (conservation strategies, titling, and governmental grants). Both aspects are hypothesized to influence deforestation, but there is little empirical evidence. We address the following questions: (i) Does timber availability attract more deforestation? (ii) Do conservation strategies (incentive-based programs in the Central Amazon and protected areas in the Chocó-Darién) influence deforestation decisions in household located outside the areas under conservation? (iii) Does the absence of titling increase the odds of a household to deforest? (iv) Can governmental grants for poverty alleviation help in the fight against deforestation? We estimated a logit model, where the dependent variable reflects whether or not a household cleared forest within the farm. As predictors, we included the above variables and controlled by household-specific characteristics. This study was conducted in the Central Amazon and the Chocó-Darién of Ecuador, two major deforestation fronts in the country. We found that timber volume potential is associated with a higher odds of deforesting in the Central Amazon, but with a lower odds in the Chocó-Darién. Although conservation strategies can influence household decisions, the effects are context-dependent. Households near the incentive-based program (Central Amazon) have a lower odds of deforesting, whereas households near a protected area (Chocó-Darién) showed the opposite effect. Titling is also important for deforestation reduction; more attention is needed in the Chocó-Darién where numerous households are living in untitled lands. Finally, governmental grants for poverty alleviation showed the potential to generate positive environmental outcomes.

THE VALUE OF THE SOLUTION OF ONE PROBLEM FOR THE PSEUDOPARABOLIC EQUATION IN THE CLASS OF GELS

This article discusses the Cauchy problem for a pseudo-parabolic equation in three-dimensional space. The result can be generalized to - dimensional space. The Cauchy problem for equations of parabolic and elliptic types is well studied. For a pseudo-parabolic equation using the previously constructed fundamental solution, evaluating the fundamental solution and its derivatives. Applying the Fourier transform with respect to and the Laplace transform with, we first obtained a priori estimates for the potentials of the initial condition and the volume potential in Hölder spaces. Further, using these results, we have proved an estimate of the solution of the Cauchy problem for the pseudo-parabolic equation in Hölder classes. A detailed proof of the estimation of the potentials of the initial condition, the volume potential, and the solution of the Cauchy problem for the pseudoparabolic equation is given

On spectral and boundary properties of the volume potential for the Helmholtz equation

In this paper, we study boundary properties and some questions of spectral geometry for certain volume potential type operators (Bessel potential operators) in an open bounded Euclidean domains. In particular, the results can be valid for differential operators, which are related to a nonlocal boundary value problem for the Helmholtz equation, so we obtain isoperimetric inequalities for its eigenvalues as well, namely, analogues of the Rayleigh-Faber-Krahn inequality.