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In this paper, we establish the equivalence of entropy and renormalized solutions of second-order elliptic equations with nonlinearities defined by the Musielak-Orlicz functions and the right-hand side from the space L1(Ω). In nonreflexive Musielak-Orlicz-Sobolev spaces, we prove the existence and uniqueness of both entropy and renormalized solutions of the Dirichlet problem in domains with a Lipschitz boundary.

A second-order quasilinear elliptic equation with a measure of special form on the right-hand side is considered. Restrictions on the structure of the equation are imposed in terms of a generalized -function such that the conjugate function obeys the -condition and the corresponding Musielak-Orlicz space is not necessarily reflexive. In an arbitrary domain satisfying the segment property, the existence of an entropy solution of the Dirichlet problem is proved. It is established that this solution is renormalized. Bibliography: 29 titles.

In this paper, we consider a certain class of anisotropic second-order elliptic equations of divergent type with variable indices of nonlinearities. We examine conditions of the solvability in the whole space ℝn, n ≥ 2. We prove the existence of solutions without restrictions to the growth rate as |x| → ∞.

The molecular mechanisms underlying the cardioprotective effect of fabomotizole were studied using the translational rat model of alcoholic cardiomyopathy developed by us. It was shown that intraperitoneal administration of fabomotizole (15 mg/kg) for 28 days to animals with alcoholic cardiomyopathy contributes to normalization of the expression of mRNA of genes of regulatory proteins СаМ (p=0.00001), Ерас1 (p=0.021), and Ерас2 (p=0.018) and receptors RyR2 (p=0.0031) and IP3R2 (p=0.006) in the myocardium of the myocardium of the left ventricle that is enhanced in control animals (p

For a class of second-order anisotropic elliptic equations with variable nonlinearity indices and summable right-hand sides, we consider the Dirichlet problem in arbitrary unbounded domains. We prove the existence and uniqueness of entropy solutions in anisotropic Sobolev spaces with variable exponents.

The mechanisms underlying cardioprotective activity of compound ALM-802 were studied in experiments on rats with chronic post-infarction heart failure. Real-time PCR showed that compound ALM-802 (daily intraperitoneal injections in a dose of 2 mg/kg for 28 days starting from day 91 after myocardial infarction modeling) restored the expression of genes encoding β1- (p=0.00001) and β2-adrenoreceptors (p=0.01) and type 2 ryanodine receptors (p=0.008) in the myocardium that was reduced in control animals. These effects can serve as the basis for the ability of the compound to reduce the intensity of remodeling and increase the inotropic function of the left heart ventricle shown earlier in this model.

A class of second-order elliptic equations with variable nonlinearity exponents and the right-hand side in the form of the general Radon measure with finite total variation is considered. The existence of a renormalized solution of the Dirichlet problem is proved as a consequence of stability with respect to the convergence of the right-hand side of the equation. Bibliography: 37 titles.

We consider a class of anisotropic elliptic equations of second order with variable exponents of non-linearity where a special Radon measure is used as the right-hand side. We establish uniqueness of entropy and renormalized solutions of the Dirichlet problem in anisotropic Sobolev spaces with variable exponents of non-linearity for arbitrary domains and certain other their properties. In addition, we prove the equivalence of entropy and renormalized solutions of the problem under consideration.

A translational rat model of chronic heart failure was employed to examine the cardioprotective effect of fabomotizole hydrochloride. Fabomotizole therapy for 28 days (15 mg/kg/day intraperitoneally) restored inotropic function of the left ventricle and increased ejection fraction from 54±3 to 65±3% (p=0.001). The inotropic function returned to normal against the background of significantly reduced myocardial expression of angiotensin (p=0.01) and glucocorticoid (p=0.03) receptors and significant increased expression of sigma-1 receptors (p=0.04). Inhibition of abnormal expression of angiotensin and glucocorticoid receptors responsible for activation of the pathological cascades underlying the postinfarction remodeling of the left ventricle as well as activation of the expression of cytoprotective sigma-1 receptors are viewed as the key features of the cardioprotective action of fabomotizole hydrochloride.

We created a translational model of chronic heart failure in rats that developed in 3 months after reproducing experimental anterior transmural myocardial infarction. The model simulated the basic clinicodiagnostic criteria of this disease: impaired contractility and dilatation of heart ventricles, signs of venous congestion, elevated plasma content of biochemical markers, and abnormal overexpression of AT1aR and β-adrenoceptors.

We consider the Dirichlet problem in an arbitrary unbounded domain with inhomogeneous boundary conditions for a certain class of anisotropic elliptic equations whose right-hand sides belong to the class L1 and prove the existence of entropic solutions in anisotropic Sobolev–Orlicz spaces.

The Dirichlet problem in arbitrary domains for a wide class of anisotropic elliptic equations of the second order with variable exponent nonlinearities and the right-hand side as a measure is consi...

The molecular mechanisms underlying the cardioprotective effect of fabomotizole were studied using the translational rat model of alcoholic cardiomyopathy developed by us. It was shown that intraperitoneal administration of fabomotizole (15 mg/kg) for 28 days to animals with alcoholic cardiomyopathy contributes to normalization of the expression of mRNA of genes of regulatory proteins СаМ (p=0.00001), Ерас1 (p=0.021), and Ерас2 (p=0.018) and receptors RyR2 (p=0.0031) and IP3R2 (p=0.006) in the myocardium of the myocardium of the left ventricle that is enhanced in control animals (p0.05). These changes were accompanied by echocardiographically documented decrease in the degree of left ventricle remodeling and improvement of its inotropic function.

Social isolation stress was modeled by long-term isolation of 12-month-old rats in individual cages over 28 weeks. It was found that sensitization of blood vessels to the vasoconstrictor action of serotonin due to overexpression of 5HT2A-type receptor genes, as well as an imbalance in the expression level of endothelin ETA- and ETB-receptors (55 and 153%, respectively) are the early signs of vascular aging. A significant contribution to the development of age-related changes in the contractile properties of blood vessels is made by the stress component, which is manifested at the level of glucocorticoid-dependent mechanisms of regulation of gene expression. The decrease in the expression of glucocorticoid receptors caused by isolation stress leads to a decrease in the expression of the genes responsible for the synthesis of V1A-R and ATII-R and to the development of vascular hyporeactivity to the vasoconstrictor action of ATII and AVP. In the aorta of stressed rats, the α1-AR mRNA level increases by 3 times. At the same time, stress did not affect the dynamics of age-related changes in the expression of genes encoding 5HT2A-R and ETA/ETB-R.

The mechanisms underlying cardioprotective activity of compound ALM-802 were studied in experiments on rats with chronic post-infarction heart failure. Real-time PCR showed that compound ALM-802 (daily intraperitoneal injections in a dose of 2 mg/kg for 28 days starting from day 91 after myocardial infarction modeling) restored the expression of genes encoding β1- (p=0.00001) and β2-adrenoreceptors (p=0.01) and type 2 ryanodine receptors (p=0.008) in the myocardium that was reduced in control animals. These effects can serve as the basis for the ability of the compound to reduce the intensity of remodeling and increase the inotropic function of the left heart ventricle shown earlier in this model.

The expression of Epac proteins (exchange protein directly activated by cAMP) and calmodulin (CaM) was assessed by the content of the corresponding mRNA in biopsy specimens of cardiac atrium, left ventricle, and thoracic aorta of rats with alcoholic cardiomyopathy. In the myocardium, overexpression of Еpac1, Ерас2, and СаМ mRNA was found. The content of Epac2 mRNA in the left ventricle was elevated by 2.9 times (p=0.000001), in the left atrium by 3.2 times (p=0.00001), in the right atrium by 3 times (p=0.00001). In contrast to the myocardial tissue, the content of CaM mRNA in the thoracic aorta was not increased, but showed a tendency to decrease, when compared to the control values, while the level of Epac1 and Epac2 mRNA was increased. The assumption is made that regulatory proteins Epac and CaM can play a key role in arrhythmogenesis development under conditions of alcoholic cardiomyopathy.

Original translational rat model of chronic heart failure provoked by experimental anterior transmural myocardium infarction was employed to examine the preventive action of anxiolytic Afobazole (15 mg/kg/day administered intraperitoneally during the first 15 days after coronary occlusion) on the development of the heart failure assessed in 3 months after infarction. Afobazole prevented the development of pathologic remodeling of the myocardium, maintained its inotropic function, and decreased the plasma level of brain natriuretic peptide known as a biochemical marker of chronic heart failure. In the myocardium, Afobazole down-regulated overexpression of the genes induced in chronic heart failure and assessed by corresponding RNA levels, which code angiotensin (AT1A-R), vasopressin (V1A-R), and glucocorticoid (GR) receptors as well as Epac2 protein. The revealed biochemical changes are consistent with the data on cardioprotective action of Afobazole.

We consider a class of anisotropic elliptic equations with nonpower nonlinearities. We establish the existence of a solution in the local Sobolev–Orlicz spaces without conditions on the data growth at infinity. We find sufficient uniqueness conditions, obtain estimates characterizing the behavior of the solution at infinity, and prove the continuous dependence of the solution on the right-hand side of the equation.

For a certain class of second-order anisotropic elliptic equations with variable nonlinearity indices and L1 right-hand side we consider the Dirichlet problem in arbitrary unbounded domains. We prove the existence and uniqueness of entropy solutions in anisotropic Sobolev spaces with variable indices.

For a certain class of anisotropic elliptic equations with the right-hand side from L1 in an arbitrary unbounded domains, the Dirichlet problem with an inhomogeneous boundary condition is considered. The existence and uniqueness of the entropy solution in anisotropic Sobolev–Orlicz spaces are proven.

Previously we have shown that at traumatic shock in rats the force of contraction of isolated aorta in response to angiotensin II, vasopressin, endothelin 1, or norepinephrine is decreased. On the contrary, vasoconstriction caused by serotonin is increased. A possible reason of the alterations of neuroendocrine regulation of vascular tone in shock may be a change in the expression of the receptors of these agonists in blood vessels. In the present study, using real-time PCR, we demonstrated that a day after injury the contents of mRNA encoding receptors V1A for vasopressin, ETA for endothelin 1, and AT1 for angiotensin II are not changed in aorta. There was a slight increase of the serotonin 5-HT2A receptor mRNA (36 ± 16%; p = 0.41). The level of the 5-HT2B receptor mRNA in aorta, initially low (2% of the content of the mRNA of receptors 5-HT2A), after the injury increased 15.8 ± 0.3 times (p < 0.01). However, at traumatic shock there was no contraction of aorta in response to 5-HT2B receptor agonist BW723C86, while vasodilation of the isolated aorta preconstricted with norepinephrine in response to BW723C86 was similar to that of the vessel isolated from control rats. The data obtained suggest that the observed 5-HT2B receptor overexpression is not related to the increased serotonin-induced vasoconstriction and might cause other vascular pathological changes at traumatic shock.

We establish the uniqueness of a solution to the Dirichlet problem for a class of anisotropic second order elliptic equations with lower-order terms and nonpower nonlinearities. We construct an example demonstrating that the class of equations under consideration is larger than the class of equations with power nonlinearites.

We found that the inhibitors of the serotonin (5HT) transporter fluoxetine and clomipramine significantly inhibit 5HT-induced constriction of isolated rings of the aorta. The most prominent inhibitory effect was observed for clomipramine, which at a concentration of 2 μM, prevented aorta constriction in response to low and moderate doses of 5HT and multiply attenuated it in response to high doses (10 μM). The inhibitors of the 5HT-transporter attenuated the strength of norepinephrine-induced aorta constriction by 40–60% and eliminated long-term tonic constriction. Application of clomipramine or fluoxetine on the vessels preliminarily constricted by norepinephrine resulted in 100% relaxation, which was maintained in the presence of the NO-synthase inhibitor L-NAME. The inhibitors of the 5HT-transporter decreased but did not prevent 5HT-induced [Ca2+]cyt increase in smooth muscle cells (SMCs) of the aorta even at high concentrations. Clomipramine and fluoxetine did not affect the vasopressin-induced [Ca2+]cyt increase in SMCs and the strength of constriction of isolated aorta rings. We found that the sensitivity of the rat aorta to the vasoconstrictor effect of 5HT and the role of the 5HT-transporter in regulation of the vascular tone increased with aging.

The paper is concerned with the solvability of the Dirichlet problem for a certain class of anisotropic elliptic second-order equations in divergence form with low-order terms and nonpolynomial nonlinearities The Caratheodory functions , , are assumed to satisfy a joint monotonicity condition in the arguments , . Constraints on their growth in are formulated in terms of a special class of convex functions. The solvability of the Dirichlet problem in unbounded domains , , is investigated. An existence theorem is proved without making any assumptions on the behaviour of the solutions and their growth as . Bibliography: 26 titles.

The paper is devoted to a certain class of doubly nonlinear higher-order anisotropic parabolic equations. Using Galerkin approximations it is proved that the first mixed problem with homogeneous Dirichlet boundary condition has a strong solution in the cylinder , where , , is an unbounded domain. When the initial function has compact support the highest possible rate of decay of this solution as is found. An upper estimate characterizing the decay of the solution is established, which is close to the lower estimate if the domain is sufficiently 'narrow'. The same authors have previously obtained results of this type for second order anisotropic parabolic equations. Bibliography: 29 titles.

Comparative study of the effect of calmodulin inhibitors trifluoperazine, W-12, and W-13 and the TRPV1 channel blocker capsazepine on receptor-dependent calcium metabolism in smooth muscle cells of the rat aorta and on the contraction of the isolated rat aorta was performed. Trifluoperazine almost completely abolishes an increase in free cytoplasmic calcium concentration in smooth muscle cells isolated from the rat aorta and smooth muscle cells of the A7r5 line in response to serotonin and does not affect cellular reaction to vasopressin and angiotensin II. W-12 and W-13 also do not attenuate responses to vasopressin and angiotensin II and reduces by two times free cytoplasmic calcium concentration elevation in response to serotonin. The efficiency of calcium metabolism suppression by calmodulin inhibitors correlates with the degree of inhibition of the aorta contractile response to serotonin. It was demonstrated that the inhibitory action of calmodulin antagonists on calcium metabolism in smooth muscle cells and the contractility of the isolated rat aorta during the activation of serotonin vasoconstrictive receptors are realized by a TRPV1-independent mechanism. It was demonstrated in experiments in vivo that trifluoperazine does not influence hypotensive reaction in rats (normally observed in response to intravenous serotonin injection), but removes the hypertensive effect of this neurotransmitter in rats after chronic introduction of dexamethasone. The results obtained confirm the hypothesis (that we previously stated) about the direct involvement of calmodulin in signal transmission from vasoconstrictive serotonin receptors.

In the paper the Dirichlet problem for an anisotropic quasilinear elliptic equations of the second order is considered. The upper estimates for the generalized solution of this Dirichlet problem are received, the closeness is proved for the isotropic case.

This work is devoted to some class of parabolic equations of high order with double nonlinearity which can be represented by a model equation ∂∂t(|u|k−2u)=∑α=1n(−1)mα−1∂mα∂xmαα[∣∣∣∂mαu∂xmαα∣∣∣pα−2∂mαu∂xmαα],m1,…,mn∈N,pn≥…≥p1>k,k>1. For the solution of the first mixed problem in a cylindrical domain D=(0,∞) ×Ω,Ω⊂Rn, n≥2, with homogeneous Dirichlet boundary condition and finite initial function the highest rate of decay established as t→∞. Earlier upper estimates were obtained by the authors for anisotropic equation of the second order and prove their accuracy.

This work is devoted to a class of parabolic equations with a double nonlinearity whose representative is a model equation (|u | k −2 u )t = n ∑ �� =1

The first initial-boundary value problem with the homogeneous Dirichlet boundary condition and a compactly supported initial function is considered for a model second-order anisotropic parabolic equation in a cylindrical domain D = (0,∞) × Ω. We find an upper bound that characterizes the dependence of the decay rate of solutions as t → ∞ on the geometry of the unbounded domain Ω ⊂ ℝn, n ≥ 3, and on nonlinearity exponents. We also obtain an estimate for the admissible decay rate of nonnegative solutions in unbounded domains; this estimate shows that the upper bound is sharp.

For a wide class of domains of revolution, we construct examples of the nonuniqueness of solutions of the first mixed problem for the heat equation, which supports the exactness of a uniqueness class of Tacklind type.

It has been shown that the agonist of 5HT1A-receptors 8-OH-DPAT induces contraction of aortic rings in the presence of angiotensin II. This effect is not associated with activation of α1-adrenoceptors by 8-OH-DPAT as it is reproduced in the presence of prazosin which completely suppresses the nonspecific vasoconstrictive effect of 8-OH-DPAT via α1-adrenoceptors on the aorta incubated without angiotensin II. Synergism in the action of angiotensin II and 8-OH-DPAT is completely preserved after partial desensitization of the receptors of angiotensin II. It has been found that 8-OH-DPAT increases the free cytoplasmic calcium concentration in cultured smooth muscle cells from the rat aorta. The data obtained support the hypothesis about the existence of “silent” vasoconstrictive 5HT1A-receptors. It has been suggested that activation of these receptors underlies synergism in vasoconstrictive action of serotonin and angiotensin II.

Agonist 5HT1A serotonin receptors 8-OH-DPAT at 70–80% in rats relax the isolated aorta and mesenteric artery, precollapsed with noradrenaline. An inhibitor of NO-synthase L-NAME two or more times suppresses vasodilatatornyh reaction in response to the effect of 8-OH-DPAT. The addition of 8-OH-DPAT to the aorta in a state of rest or precollapsed with endothelin-1 or vasopressin causes an increase in apower reduction. A blocker of α1-adrenoceptors prazosin almost completely suppresses the aorta collapse reaction to the effect of 8-OH-DPAT in the absence of vasoconstrictives, but does not affect the contraction force in response to 8-OH-DPAT of the aorta in the presence of endothelin-1 or vasopressin and does not shift the curve of the dependence of force collapse on the concentration of 8-OH-DPAT. Our data show the existence in the rat aorta of vasodilator and vasoconstrictive 5HT1A-receptors.

Upper bounds are obtained for solutions of the Dirichlet problem for pseudodifferential elliptic equations where the right-hand side has compact support. In domains with non-compact boundary they characterise the behaviour of solutions at infinity in its dependence on the geometric properties of the domain. For unbounded domains where the boundary has irregular behaviour, it is shown that these bounds may be more efficient than the bounds that are already known for second-order elliptic equations. For second-order elliptic equations in a broad class of domains of revolution these bounds are shown to be sharp.Bibliography: 17 titles.

In a cylindric domain D = (0, ∞) × Ω where Ω ⊂ ℝn+1 is an unbounded domain, the first mixed problem for a high-order parabolic equation $$u_t + ( - 1)^k D_x^k (a(x,y)D_x^k u) + \sum\limits_{i = l}^m {\sum\limits_{\left| \alpha \right| = \left| \beta \right| = i} {( - 1)^i D_y^\alpha } } (b_{\alpha \beta } (x,y)D_y^\beta u) = 0, l \leqslant m, k,l,m \in \mathbb{N},$$ is considered. The boundary values are homogeneous and the initial value is a finite function. In terms of the new geometrical characteristics of the domain, the upper estimate of L2-norm ∥u(t)∥ of the solution to the problem is established. In particular, in domains {(x, y) ∈ ℝn+1 | x > 0, |y1| < xa}, 0 < a < q/l, under the assumption that the upper and lower symbols of the operator L are separated from zero, this estimate takes the form Open image in new window . This estimate is determined by minor terms of the equation. The sharpness of the estimate for a wide class of unbounded domains is proved in the case k = l = m = 1.

Normally serotonin reduced blood pressure. It was shown that in rats with traumatic shock its hypotensive effect was transformed into hypertensive one. In vitro serotonin exhibited a slight vasoconstrictor effect on isolated rat aorta, while 24 h after injury, the strength of aortic contractions in response to serotonin increased 2.2 times. Desensitization of glucocorticoid receptors caused by injection of high doses of dexamethasone (3 mg/kg) to rats for 5 days led to similar changes in serotonin effect. We hypothesized that inversion of the response to serotonin in shock was caused by increased activity and/or expression of vasoconstrictor serotonin receptors in blood vessels.

Membrane receptors are known to be targets of the majority of modern drugs. However, alterations of these receptors in diseases, causes of these alterations, and their relations with the development of diseases are studied only in few cases. This review considers some mechanisms of regulation of activity and expression of G-protein-coupled receptors (GPCRs) that comprise the largest family of membrane proteins. Data are presented on hereditary diseases caused by genetic deficiencies of receptors, and on oncological diseases caused by somatic mutations of GPCR(s) and G-protein(s) genes. The relation is discussed between the development of diseases and disorders in the receptor functioning that appear during ontogenesis. In the final part of the review our own data are presented on the state of the receptors of vasoconstrictive hormones and neurotransmitters in blood vessels under the conditions of traumatic shock.

In a cylindrical domain , where is an unbounded subdomain of , one considers the evolution equation the right-hand side of which is a quasi-elliptic operator with highest derivatives of orders with respect to the variables . For the mixed problem with Dirichlet condition at the lateral part of the boundary of? a uniqueness class of the T?cklind type is described.For domains tapering at infinity another uniqueness class is distinguished, a geometric one, which is broader than the T?cklind-type class. It is shown that for domains with irregular behaviour of the boundary this class is wider than the one described for a second-order parabolic equation by Ole?nik and Iosif'yan (Uspekhi Mat. Nauk, 1976?[17]). In a wide class of tapering domains non-uniqueness examples for solutions of the first mixed problem for the heat equation are constructed, which supports the exactness of the geometric uniqueness class.Bibliography: 33 titles.

We investigated the role of 5HT2C receptors in regulation of blood vessel contractility. We determined expression of 5HT2C receptors in smooth muscle cell line A7r5 as well as on isolated rat aorta. It was shown that strong vasoconstriction effect of 5HT2C receptor agonists - SCH 23390 and MK 212 appeared on blood vessels after preliminary activation of angiotensin ATIA- and vasopressin V1A-receptors. Biphasic contraction (a rhythmic alternation of contraction and subsequent relaxation phases of aoitic rings) and tonic contraction were observed in 75% and 25% of the cases after 5HT2C receptor activation, respectively. Periodic high amplitude constrictions of isolated rat aorta, induced by SCH 23390 and MK 212 agonists, were persisted for a long time (1 hour). It was revealed that calmodulin and c-Src kinase play a central role in the mechanisms of signal transduction from 5HT2C receptors. Trifluoperazine and PP2, the inhibitors of calmodulin and c-Src kinase, respectively, abolished vasoconstriction reaction of isolated aortic rings in response to SCH 23390 and MK 212 but did not affect the strength gain of the vasoconstriction caused by fluoroaluminate, a G-protein activator. Taken together, these date suggest that 5HT2C receptors are in a latent state in blood vessels (silentreceptors) and activation of these receptors is dependent on the functional state of the receptors of other endogenous vasoconstrictors.

We found that the inhibitor of Rho-kinase fasudil selectively inhibited constriction of isolated rings of the aorta and mesenteric artery in rats in response to application of the agonists of 5HT2A-(DOI and TBC-2) and 5HT1A-receptors (8-OH-DPAT) and did not influence vasoconstriction induced by serotonin. We demonstrate for the first time that application of the agonists of 5HT2C-receptors (MK 212 and SCH 23390) did not influence the tone of "intact" vessels. The marked vasoconstrictory effect of the agonists of 5HT2C-receptors was observed in the vessels preconstricted due to angiotensin II or vasopressin. We found that the inhibitor of Rho-kinase did not influence negatively on MK 212 or SCH 23390-induced constriction of isolated rings of the aorta and mesenteric artery in rats. We suppose.that, in the presence of fasudil, serotonin induces constriction of vessels through the interaction with 5HT2C-receptors and signal transduction from these receptors does not involve Rho-kinase activity. We found that fasudil attenuated vasoconstriction induced by norepinephrine and vasopressin by 40%. We.demonstrated that tyrosine c-Src-kinase plays the most important role in signal transduction from 5HT-receptors because its effects are specific with relation to these receptors.

We select a class of uniqueness of the solutions of the quasi-elliptic equation with the Dirichlet condition on the boundary of an unbounded domain and show that for domains with irregular behaviour of the boundary this class can be wider than that established in [10] for second-order elliptic equations. For the Laplace equation we construct an example of non-uniqueness of solution of the Dirichlet problem that shows that the class of uniqueness found in this paper cannot be essentially extended.

In a cylindrical domain , where is an unbounded subdomain of , one considers the first mixed problem for a higher order equation with homogeneous boundary conditions and compactly supported initial function. A new method of obtaining an upper estimate of the -norm of the solution of this problem is put forward, which works in a broad class of domains and equations. In particular, in domains , , for the operator with symbol satisfying a certain condition this estimate takes the following form: The estimate is shown to be sharp in a broad class of unbounded domains for , that is, for second-order parabolic equations.