Phase Transition and Coefficients of Thermal Expansion in Al2−xInxW3O12 (0.2 ≤ x ≤ 1)

Materials from theA₂M₃O₁₂ family are known for their extensive chemical versatility while preserving the polyhedral-corner-shared orthorhombic crystal system, as well as for their consequent unusual thermal expansion, varying from negative and near-zero to slightly positive. The rarest are near-zero thermal expansion materials, which are of paramount importance in thermal shock resistance applications. Ceramic materials with chemistry Al_2−xIn_xW₃O₁₂ (x = 0.2–1.0) were synthesized using a modified reverse-strike co-precipitation method and prepared into solid specimens using traditional ceramic sintering. The resulting materials were characterized by X-ray powder diffraction (ambient and in situ high temperatures), differential scanning calorimetry and dilatometry to delineate thermal expansion, phase transitions and crystal structures. It was found that the x = 0.2 composition had the lowest thermal expansion, 1.88 × 10⁻⁶ K⁻¹, which was still higher than the end member Al₂W₃O₁₂ for the chemical series. Furthermore, the AlInW₃O₁₂ was monoclinic phase at room temperature and transformed to the orthorhombic form at ca. 200 °C, in contrast with previous reports. Interestingly, the x = 0.2, x = 0.4 and x = 0.7 materials did not exhibit the expected orthorhombic-to-monoclinic phase transition as observed for the other compositions, and hence did not follow the expected Vegard-like relationship associated with the electronegativity rule. Overall, compositions within the Al_2−xIn_xW₃O₁₂ family should not be considered candidates for high thermal shock applications that would require near-zero thermal expansion properties.     

Crystalline Quality,Composition Homogeneity,Tellurium Precipitates/Inclusions Concentration,Optical Transmission,and Energy Band Gap of Bridgman Grown Single-Crystalline Cd1−xZnxTe (0 ≤ x ≤ 0.1)

Cd_1−xZn_xTe (0 ≤ x ≤ 0.1) ingots were obtained by Bridgman’s method using two different speeds in order to find the optimal conditions for single-crystalline growth. Crystalline quality was studied by chemical etching, the elemental composition by wavelength dispersive spectroscopy (WDS), tellurium (Te) precipitates/inclusions concentration by differential scanning calorimetry (DSC), optical transmission by Fourier transformed infrared spectrometry (FTIR), and band gap energy (Egap) by photoluminescence (PL). It was observed that the ingots grown at a lower speed were those of the best crystalline quality, having at most three grains of different crystallographic orientation. The average dislocations density in all of them were similar and correspond to materials of good quality. EPMA results indicated that the homogeneity in the composition was excellent in the ingots central part. The concentration of Te precipitates/inclusions in all ingots was below the instrument (DSC) detection limit, 0.25% wt/wt. In the case of wafers from Cd_0.96Zn_0.04Te and Cd_0.90Zn_0.10Te ingots, the optical transmission was better than that of commercial materials and varied between 60% and 70%, while for pure CdTe, the transmission range was between 50% and 55%, the latter being decreased by the presence of Te precipitates/inclusions. The band gap energy Eg of different wafers was experimentally obtained by PL measurements at 76 K. We observed that Eg increased with the Zn concentration of the wafers, following a linear regression comparable to those proposed in the literature, and consistent with the results obtained with other techniques.     

Structural,Magnetic, Dielectric and Piezoelectric Properties of Multiferroic PbTi1−xFexO3−δ Ceramics

PbTi_1−xFe_xO_3−δ (x = 0, 0.3, 0.5, and 0.7) ceramics were prepared using the classical solid-state reaction method. The investigated system presented properties that were derived from composition, microstructure, and oxygen deficiency. The phase investigations indicated that all of the samples were well crystallized, and the formation of a cubic structure with small traces of impurities was promoted, in addition to a tetragonal structure, as Fe³⁺ concentration increased. The scanning electron microscopy (SEM) images for PbTi_1−xFe_xO_3−δ ceramics revealed microstructures that were inhomogeneous with an intergranular porosity. The dielectric permittivity increased systematically with Fe³⁺ concentration, increasing up to x = 0.7. A complex impedance analysis revealed the presence of multiple semicircles in the spectra, demonstrating a local electrical inhomogeneity due the different microstructures and amounts of oxygen vacancies distributed within the sample. The increase of the substitution with Fe³⁺ ions onto Ti⁴⁺ sites led to the improvement of the magnetic properties due to the gradual increase in the interactions between Fe³⁺ ions, which were mediated by the presence of oxygen vacancies. The PbTi_1−xFe_xO_3−δ became a multifunctional system with reasonable dielectric, piezoelectric, and magnetic characteristics, making it suitable for application in magnetoelectric devices.     

Polar Order Evolutions near the Rhombohedral to Pseudocubic and Tetragonal to Pseudocubic Phase Boundaries of the BiFeO3-BaTiO3 System

Solid solutions of (1-x)BiFeO₃-xBaTiO₃ (BF-BT, 0.05 ≤ x ≤ 0.98) were prepared and characterized. It was found that the dielectric constant ε_m, remnant polarization P_r and piezoelectric coefficient d₃₃ reach their maximum values near the rhombohedral–pseudocubic phase boundary. In particular, the 0.7BF-0.3BT composition shows large polarization (P_r > 20 μC/cm²) and a temperature-stable piezoelectric property (d₃₃ > 100 pC/N when the annealing temperature is lower than ~400 °C). Near the tetragonal–pseudocubic phase boundary, ε_m and P_r decrease, and the piezoelectric property vanishes when the BF content reaches 4 mol %.     

The Dielectric Constant of Ba6−3x(Sm1−yNdy)8+2xTi18O54 (x = 2/3) Ceramics for Microwave Communication by Linear Regression Analysis

The electronics related to the fifth generation mobile communication technology (5G) are projected to possess significant market potential. High dielectric constant microwave ceramics used as filters and resonators in 5G have thus attracted great attention. The Ba_6−3x(Sm_1−yNd_y)_8+2xTi₁₈O₅₄ (x = 2/3) ceramic system has aroused people’s interest due to its underlying excellent microwave dielectric properties. In this paper, the relationships between the dielectric constant, Nd-doped content, sintering temperature and the density of Ba_6−3x(Sm_1−yNd_y)_8+2xTi₁₈O₅₄ (x = 2/3) ceramics were studied. The linear regression equation was established by statistical product and service solution (SPSS) data analysis software, and the factors affecting the dielectric constant have been analyzed by using the enter and stepwise methods, respectively. It is found that the model established by the stepwise method is practically significant with Y = −71.168 + 6.946x₁ + 25.799x₃, where Y, x₁ and x₃ represent the dielectric constant, Nd content and the density, respectively. According to this model, the influence of density on the dielectric constant is greater than that of Nd doping concentration. We bring the linear regression analysis method into the research field of microwave dielectric ceramics, hoping to provide an instructive for the optimization of ceramic technology.     

New Solid Solution and Phase Equilibria in the Subsolidus Area of the Three-Component CuO–V2O5–Ta2O5 Oxide System

The results of the study of the three-component system of CuO–V₂O₅–Ta₂O₅ oxides showed, inter alia, that in the air atmosphere in one of its cross-sections, i.e., in the CuV₂O₆–CuTa₂O₆ system, a new substitutional solid solution with the general formula CuTa_2−xV_xO₆ and homogeneity range for x > 0.0 and x ≤ 0.3 is formed. The influence of the degree of incorporation of V⁵⁺ ions into the CuTa₂O₆ crystal lattice in place of Ta⁵⁺ ions on the unit cell volume, thermal stability and IR spectra of the obtained solid solution was determined. Moreover, the value of the band gap energy of the CuTa_2−xV_xO₆ solid solution was estimated in the range of 0.0 < x ≤ 0.3, and on this basis, the new solid solution was classified as a semiconductor. On the basis of the research results, the studied system of CuO–V₂O₅–Ta₂O₅ oxides was also divided into 12 subsidiary subsystems.     

Athermal ω Phase and Lattice Modulation in Binary Zr-Nb Alloys

To further explore the potential of Zr-based alloys as a biomaterial that will not interfere with magnetic resonance imaging (MRI), the microstructural characteristics of Zr-xat.% Nb alloys (10 ≤ x ≤ 18), particularly the athermal ω phase and lattice modulation, were investigated by conducting electrical resistivity and magnetic susceptibility measurements and transmission electron microscopy observations. The 10 Nb alloy and 12 Nb alloys had a positive temperature coefficient of electrical resistivity. The athermal ω phase existed in 10 Nb and 12 Nb alloys at room temperature. Alternatively, the 14 Nb and 18 Nb alloys had an anomalous negative temperature coefficient of the resistivity. The selected area diffraction pattern of the 14 Nb alloy revealed the co-occurrence of ω phase diffraction and diffuse satellites. These diffuse satellites were represented by g_β + q when the zone axis was [001] or [113], but not [110]. These results imply that these diffuse satellites appeared because the transverse waves consistent with the propagation and displacement vectors were q = <ζ ζ¯ 0>* for the ζ~1/2 and <110> directions. It is possible that the resistivity anomaly was caused by the formation of the athermal ω phase and transverse wave. Moreover, control of the athermal ω-phase transformation and occurrence of lattice modulation led to reduced magnetic susceptibility, superior deformation properties, and a low Young’s modulus in the Zr-Nb alloys. Thus, Zr-Nb alloys are promising MRI-compatible metallic biomaterials.     

Growth and Investigation of Annealing Effects on Ternary Cd1−xMgxO Nanocomposit Thin Films

Thin films of Cd_1−xMg_xO (CdMgO) (0 ≤ x ≤ 1) were investigated by depositing the films on glass substrates using the co-evaporation technique. The structural, surface morphological, optical, and electrical characteristics of these films were studied as a function of Mg content after annealing at 350 °C. The XRD analysis showed that the deposited films had an amorphous nature. The grain size of the films reduced as the Mg concentration increased, as evidenced by the surface morphology, and EDAX supported the existence of Mg content. It was observed that as the films were annealed, the transmittance of the CdMgO films saw an increase of up to 85%. The blue shift of the absorption edge was observed by the increase of Mg content, which was useful for enhancing the efficiency of solar cells. The optical band gap increased from 2.45 to 6.02 eV as the Mg content increased. With increased Mg content, the refractive index reduced from 2.49 to 1.735, and electrical resistivity increased from 535 Ω cm to 1.57 × 10⁶ Ω cm.     

Metastable Materials Accessed under Moderate Pressure Conditions (P ≤ 3.5 GPa) in a Piston-Cylinder Press

In this review, we describe different families of metastable materials, some of them with relevant technological applications, which can be stabilized at moderate pressures 2–3.5 GPa in a piston-cylinder press. The synthesis of some of these systems had been previously reported under higher hydrostatic pressures (6–10 GPa), but can be accessed under milder conditions in combination with reactive precursors prepared by soft-chemistry techniques. These systems include perovskites with transition metals in unusual oxidation states (e.g., RNiO₃ with Ni³⁺, R = rare earths); double perovskites such as RCu₃Mn₄O₁₂ with Jahn–Teller Cu²⁺ ions at A sites, pyrochlores derived from Tl₂Mn₂O₇ with colossal magnetoresistance, pnictide skutterudites M_xCo₄Sb₁₂ (M = La, Yb, Ce, Sr, K) with thermoelectric properties, or metal hydrides Mg₂MH_x (M = Fe, Co, Ni) and AMgH₃ (A: alkali metals) with applications in hydrogen storage. The availability of substantial amounts of sample (0.5–1.5 g) allows a complete characterization of the properties of interest, including magnetic, transport, thermoelectric properties and so on, and the structural characterization by neutron or synchrotron X-ray diffraction techniques.     

Corrosion Behavior of CrFeCoNiVx (x = 0.5 and 1) High-Entropy Alloys in 1M Sulfuric Acid and 1M Hydrochloric Acid Solutions

CrFeCoNiV_x high-entropy alloys were prepared by arc-melting, and the microstructures and corrosion properties of these alloys were studied. The CrFeCoNiV_0.5 alloy had a granular structure; the matrix was a face-centered cubic (FCC) structure, and the second phase was a σ phase with a tetragonal structure. The CrFeCoNiV alloy had a dendritic structure; the dendrites in this alloy showed an FCC phase, and the interdendrities had a eutectic structure of FCC and σ phases. Therefore, CrFeCoNiV was much harder than the CrFeCoNiV_0.5 alloy due to the dendritic structures. The potentiodynamic polarization test and electrochemical impedance spectroscopy were used to evaluate the corrosion behavior of the CrFeCoNiV_x high-entropy alloys in deaerated 1M sulfuric acid and 1M hydrochloric acid solutions. The results indicated that the CrFeCoNiV_0.5 alloy had a better corrosion resistance because of the granular structure.     

An A- and B-Site Substitutional Study of SrFeO3−δ Perovskites for Solar Thermochemical Air Separation

An A‑ and B‑site substitutional study of SrFeO_3−δ perovskites (A’_xA_1−xB’_yB_1−yO_3−δ, where A = Sr and B = Fe) was performed for a two‑step solar thermochemical air separation cycle. The cycle steps encompass (1) the thermal reduction of A’_xSr_1−xB’_yFe_1−yO_3−δ driven by concentrated solar irradiation and (2) the oxidation of A’_xSr_1−xB’_yFe_1−yO_3−δ in air to remove O₂, leaving N₂. The oxidized A’_xSr_1−xB’_yFe_1−yO_3−δ is recycled back to the first step to complete the cycle, resulting in the separation of N₂ from air and concentrated solar irradiation. A-site substitution fractions between 0 ≤ x ≤ 0.2 were examined for A’ = Ba, Ca, and La. B-site substitution fractions between 0 ≤ y ≤ 0.2 were examined for B’ = Cr, Cu, Co, and Mn. Samples were prepared with a modified Pechini method and characterized with X-ray diffractometry. The mass changes and deviations from stoichiometry were evaluated with thermogravimetry in three screenings with temperature- and O₂ pressure-swings between 573 and 1473 K and 20% O₂/Ar and 100% Ar at 1 bar, respectively. A’ = Ba or La and B’ = Co resulted in the most improved redox capacities amongst temperature- and O₂ pressure-swing experiments.     

Electrical and Structural Properties of Li1.3Al0.3Ti1.7(PO4)3—Based Ceramics Prepared with the Addition of Li4SiO4

The currently studied materials considered as potential candidates to be solid electrolytes for Li-ion batteries usually suffer from low total ionic conductivity. One of them, the NASICON-type ceramic of the chemical formula Li_1.3Al_0.3Ti_1.7(PO₄)₃, seems to be an appropriate material for the modification of its electrical properties due to its high bulk ionic conductivity of the order of 10⁻³ S∙cm⁻¹. For this purpose, we propose an approach concerning modifying the grain boundary composition towards the higher conducting one. To achieve this goal, Li₄SiO₄ was selected and added to the LATP base matrix to support Li⁺ diffusion between the grains. The properties of the Li_1.3Al_0.3Ti_1.7(PO₄)₃−xLi₄SiO₄ (0.02 ≤ x ≤ 0.1) system were studied by means of high-temperature X-ray diffractometry (HTXRD); ⁶Li, ²⁷Al, ²⁹Si, and ³¹P magic angle spinning nuclear magnetic resonance spectroscopy (MAS NMR); thermogravimetry (TG); scanning electron microscopy (SEM); and impedance spectroscopy (IS) techniques. Referring to the experimental results, the Li₄SiO₄ additive material leads to the improvement of the electrical properties and the value of the total ionic conductivity exceeds 10⁻⁴ S∙cm⁻¹ in most studied cases. The factors affecting the enhancement of the total ionic conductivity are discussed. The highest value of σ_tot = 1.4 × 10⁻⁴ S∙cm⁻¹ has been obtained for LATP–0.1LSO material sintered at 1000 °C for 6 h.     

Structural and Optical Properties of La1?xSrxTiO3+δ

La_1−xSr_xTiO_3+δ has attracted much attention as an important perovskite oxide. However, there are rare reports on its optical properties, especially reflectivity. In this paper, its structural and optical properties were studied. The X-ray diffraction, X-ray photoelectron spectroscopy, scanning electron microscopy and spectrophotometer were used to characterize the sample. The results show that with increasing Sr concentration, the number of TiO₆ octahedral layers in each “slab” increases and the crystal structure changes from layered to cubic structure. A proper Sr doping (x = 0.1) can increase the reflectivity, reaching 95% in the near infrared range, which is comparable with metal Al measured in the same condition. This indicates its potential applications as optical protective coatings or anti-radiation materials at high temperatures.     

Study on Mechanical Behavior and Energy Mechanism of Sandstone under Chemical Corrosion

Chemical corrosion has a significant impact on the properties of rock materials. To study the mechanical behavior and energy mechanism of rock under chemical corrosion, this paper took the sandstone of Haitangshan tunnel in Fuxin as the research object, used a Na₂SO₄ solution to simulate different chemical environments, carried out a triaxial loading test on sandstone through the MTS815.02 test system, and analyzed the mechanical parameters and energy damage evolution law of sandstone under different chemical environments. The test results showed that the basic mechanical parameters (peak strength σ_pk, peak strain ε_pk, elastic modulus E, cohesion c, and internal friction angle φ) and characteristic stress parameters (closure stress σ_cc, initiation stress σ_ci, and dilatancy stress σ_cd) of sandstone first increased and then decreased with the increase of pH in the Na₂SO₄ solution, Poisson’s ratio µ showed the opposite trend, and the extreme values of all parameters were taken when pH = 7. The influence degree of different pHs on the mechanical parameters of sandstone were as follows: strong acid environment (pH ≤ 4) > strong alkali environment (pH ≥ 10) > weak acid environment (4 ≤ pH < 6) > weak alkali environment (8 ≤ pH < 10) > neutral environment (6 < pH< 8). The total energy and elastic strain energy increased first and then decreased, and the dissipated energy was the opposite. The damage variable decreased first and then increased. With the increasing concentration of the Na₂SO₄ solution, all the above parameters changed monotonically. Based on the energy theory, the damage evolution equation considering the effect of the Na₂SO₄ concentration was established. Combined with the test data, the model was verified and the result was good. Under the action of Na₂SO₄ corrosion, Ca²⁺ in calcite and Fe²⁺ in hematite were dissolved and precipitated. With the gradual increase of Ca²⁺ and Fe²⁺ concentration, the damage variable increased gradually. The relationship between the two ion concentrations and the damage variable approximately satisfied a linear function.     

Crystal and Magnetic Structure Transitions in BiMnO3+δ Ceramics Driven by Cation Vacancies and Temperature

The crystal structure of BiMnO_3+δ ceramics has been studied as a function of nominal oxygen excess and temperature using synchrotron and neutron powder diffraction, magnetometry and differential scanning calorimetry. Increase in oxygen excess leads to the structural transformations from the monoclinic structure (C2/c) to another monoclinic (P2₁/c), and then to the orthorhombic (Pnma) structure through the two-phase regions. The sequence of the structural transformations is accompanied by a modification of the orbital ordering followed by its disruption. Modification of the orbital order leads to a rearrangement of the magnetic structure of the compounds from the long-range ferromagnetic to a mixed magnetic state with antiferromagnetic clusters coexistent in a ferromagnetic matrix followed by a frustration of the long-range magnetic order. Temperature increase causes the structural transition to the nonpolar orthorhombic phase regardless of the structural state at room temperature; the orbital order is destroyed in compounds BiMnO_3+δ (δ ≤ 0.14) at temperatures above 470 °C.     

Study of the Structure,Magnetic, Thermal and Electrical Characterisation of ZnCr2Se4: Ta Single Crystals Obtained by Chemical Vapour Transport

The new series of single-crystalline chromium selenides, Ta-doped ZnCr₂Se₄, was synthesised by a chemical vapour transport method to determine the impact of a dopant on the structural and thermodynamic properties of the parent compound. We present comprehensive investigations of structural, electrical transport, magnetic, and specific heat properties. It was expected that a partial replacement of Cr ions by a more significant Ta one would lead to a change in direct magnetic interactions between Cr magnetic moments and result in a change in the magnetic ground state and electric transport properties of the ZnCr_2−xTa_xSe₄ (x = 0.05, 0.06, 0.07, 0.08, 0.1, 0.12) system. We found that all the elements of the cubic system had a cubic spinel structure; however, the doping gain linearly increased the ZnCr_2−xTa_xSe₄ unit cell volume. Doping with tantalum did not significantly change the semiconductor and magnetic properties of ZnCr₂Se₄. For all studied samples (0 ≤ x ≤ 0.12), an antiferromagnetic order (AFM) below T_N~22 K was observed. However, a small amount of Ta significantly reduced the second critical field (H_c2) from 65 kOe for x = 0.0 (ZnCr₂Se₄ matrix) up to 42.2 kOe for x = 0.12, above which the spin helical system changed to ferromagnetic (FM). The H_c2 reduction can lead to strong competition among AFM and FM interactions and spin frustration, as the specific heat under magnetic fields H < H_c2 shows a strong field decrease in T_N.     

Enhancing Thermoelectric Properties of (Cu2Te)1−x-(BiCuTeO)x Composites by Optimizing Carrier Concentration

Because of the high carrier concentration, copper telluride (Cu₂Te) has a relatively low Seebeck coefficient and high thermal conductivity, which are not good for its thermoelectric performance. To simultaneously optimize carrier concentration, lower thermal conductivity and improve the stability, BiCuTeO, an oxygen containing compound with lower carrier concentration, is in situ formed in Cu₂Te by a method of combining self-propagating high-temperature synthesis (SHS) with spark plasma sintering (SPS). With the incorporation of BiCuTeO, the carrier concentration decreased from 8.1 × 10²⁰ to 3.8 × 10²⁰ cm⁻³, bringing the increase of power factor from ~1.91 to ~2.97 μW cm⁻¹ K⁻² at normal temperature. At the same time, thermal conductivity reduced from 2.61 to 1.48 W m⁻¹ K⁻¹ at 623 K. Consequently, (Cu₂Te)_0.95-(BiCuTeO)_0.05 composite sample reached a relatively high ZT value of 0.13 at 723 K, which is 41% higher than that of Cu₂Te.     

Formation of Solid Solutions and Physicochemical Properties of the High-Entropy Ln1−xSrx(Co,Cr,Fe,Mn,Ni)O3−δ (Ln = La,Pr, Nd,Sm or Gd) Perovskites

Phase composition, crystal structure, and selected physicochemical properties of the high entropy Ln(Co,Cr,Fe,Mn,Ni)O_3−δ (Ln = La, Pr, Gd, Nd, Sm) perovskites, as well as the possibility of Sr doping in Ln_1−xSr_x(Co,Cr,Fe,Mn,Ni)O_3−δ series, are reported is this work. With the use of the Pechini method, all undoped compositions are successfully synthesized. The samples exhibit distorted, orthorhombic or rhombohedral crystal structure, and a linear correlation is observed between the ionic radius of Ln and the value of the quasi-cubic perovskite lattice constant. The oxides show moderate thermal expansion, with a lack of visible contribution from the chemical expansion effect. Temperature-dependent values of the total electrical conductivity are reported, and the observed behavior appears distinctive from that of non-high entropy transition metal-based perovskites, beyond the expectations based on the rule-of-mixtures. In terms of formation of solid solutions in Sr-doped Ln_1−xSr_x(Co,Cr,Fe,Mn,Ni)O_3−δ materials, the results indicate a strong influence of the Ln radius, and while for La-based series the Sr solubility limit is at the level of x_max = 0.3, for the smaller Pr it is equal to just 0.1. In the case of Nd-, Sm- and Gd-based materials, even for the x_Sr = 0.1, the formation of secondary phases is observed on the SEM + EDS images.     

Structure and Phase State of Ti49.4Ni50.6 (at%) Hydrogenated in Normal Saline

The paper analyzes the surface structure and phase state of Ti_49.4Ni_50.6 (at%) hydrogenated at 295 K in normal saline (0.9% NaCl aqueous solution with pH = 5.7) at 20 A/m² for 0.5–6 h. The analysis shows that the average hydrogen concentration in the alloy increases with the hydrogenation time t_H as follows: slowly to 50 ppm at t_H = 0.5–1.5 h, steeply to 150 ppm at t_H = 1.2–2 h, and linearly to 300 ppm at t_H = 2–6 h. According to Bragg–Brentano X-ray diffraction data (θ–2 θ, 2 θ ≤ 50°, CoKα radiation), the alloy in its scanned surface layer of thickness ~5.6 µm reveals a TiNiH_x phase with x = 0.64 and x = 0.54 after hydrogenation for 4 and 6 h, respectively. The structure of this phase is identifiable as an orthorhombic hydride similar to β₁–TiFeH_0.94 (space group Pmcm), rather than as a tetragonal TiNiH_x hydride with x = 0.30–1.0 (space group I4/mmm). Time curves are presented to trace the lattice parameters and volume change during the formation of such an orthorhombic phase from the initial cubic B2 phase in Ti_49.4Ni_50.6 (at%).     

Blow-up of solutions of nonlinear wave equations in three space dimensions

Solutions u(x, t) of the inequality □u ≥ A|u|^p for x ε R₃, t ≥ 0 are considered, where □ is the d'Alembertian, and A,p are constants with A > 0, 1 < p < 1 + √2. It is shown that the support of u is compact and contained in the cone 0 ≤ t ≤ t₀ -|x - x⁰|, if the “initial data” u(x, 0), u_t(x, 0) have their support in the ball|x - x⁰| ≤ t₀. In particular, “global” solutions of □u = A|u|^p with initial data of compact support vanish identically. On the other hand, for A > 0, p > 1 + √2, global solutions of □u = A|u|^p exist, if the initial data are of compact support and “sufficiently” small.