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In this article, we investigate the Cauchy problem for Klein-Gordon equations with combined power-type nonlinearities. Coefficients in the nonlinearities depend on the space variable. They are sign preserving functions except one of the coefficients, which may change its sign. We study completely the structure of the Nehari manifold. By using the potential well method, we give necessary and sufficient conditions for nonexistence of global solution for subcritical initial energy by means of the sign of the Nehari functional. When the energy is positive, we propose new sufficient conditions for finite time blow up of the weak solutions. One of these conditions is independent of the sign of the scalar product of the initial data. We also prove uniqueness of the weak solutions under slightly more restrictive assumptions for the powers of the nonlinearities.

Cyclic organochlorine compounds, including hexachlorobenzene (HCB) being among the most dangerous persistent organic pollutants, have accumulated in the environment due to their widespread use for chemical plant protection. The issue of different HCB contamination influence on soil microbial communities is relevant and insufficiently studied. The aim was to investigate the reaction of microbial communities to HCB contamination of chernozem, dark-kastanozem and sod-podzolic soils of Ukraine. Methods. Model laboratory experiments, microbiological (determination of the microbial quantity on agar nutrient media), chemical (adsorption method for the study of basal and substrate-induced respiration of soil microbiota), and statistical methods. Results. Contamination of chernozem, dark-kastanozem and sod-podzolic soils of Ukraine with HCB in doses from 10 to 10,000 maximum permissible concentrations adversely affected the microbial quantity of major ecological and trophic groups, among which the most sensitive were phosphate-mobilizing bacteria and streptomycetes. In contaminated soils, the structure of the microbiocenoses was disturbed, the general biological activity was reduced - the basal respiration rate by 28.5-62.7% and the substrate-induced by 2-3 times, the accumulation of microbial biomass decreased by 1.5-4.3 times. The HCB contamination caused the main impact on the dispersion of microbial quantity (by 61-95%), the influence of soil type was smaller (1-24%). Conclusions. Microbiocenoses of chernozem, dark-kastanozem, sod-podzolic soils in intensive land use systems are vulnerable to HCB contamination in doses from 10 to 10.000 maximum permissible concentrations. The most sensitive phosphate-mobilizing bacteria and streptomycetes can be used as indicators in monitoring of organochlorine contaminated soils. Under the action of pesticide loading, microbial respiration and accumulation of microbial biomass are suppressed. The negative reaction of microbial communities to HCB contamination indicates the need for remediation measures to recovery the microbiota and soil fertility. [ABSTRACT FROM AUTHOR]

Many research has been devoted to the influence of organochlorine pesticides (OCP) on soil microorganisms, but there is little information available on effect of different levels of contamination with hexachlorobenzene (HCB) on soil microbiota. The purpose of this study was to investigate the influence of various doses of HCB on microbiota of chernozem soil. Methods. Quantity of microorganisms was assessed using the method of sowing tenfold dilutions of the soil suspension on agarified nutrient media and assessing the number of colony-forming units (CFUs). The results of the research showed that the HCB effect on microbial quantity of functional groups and their ratio was negative. The most susceptible to HCB action were cellulolytic fungi and streptomycetes, the most stable ones − soil micromycetes. Conclusions. HCB effect disrupts both the mechanisms as resistance and flexible stability of microbial communities. [ABSTRACT FROM AUTHOR]

The global behaviour of the weak solutions of the Cauchy problem to nonlinear Klein–Gordon equation with combined power-type nonlinearity is studied. Finite time blow-up of the solutions with arbitrary high positive initial energy is proved under general structural conditions on the initial data. A new functional, invariant under the flow of the equation, is introduced and investigated. [ABSTRACT FROM AUTHOR]