Nedyu Popivanov

Sofia University "St. Kliment Ohridski" · Department of Differential Equations

In the 1950s, Protter proposed multi-dimensional analogues of the classical Guderley–Morawetz problem for mixed-type hyperbolic-elliptic equations on the plane that models transonic flows in fluid dynamics. The multi-dimensional variants turn out to be different from the two-dimensional case and the situation there is still not clear. Here, we stud...

Bulgaria has the lowest COVID-19 vaccination rate in the European Union and the second-highest COVID-19 mortality rate in the world. That is why we think it is important better to understand the reason for this situation and to analyse the development of the disease over time. In this paper, an extended time-dependent SEIRS model SEIRS-VB is used t...

In this paper some ill-posed boundary-value problems (BVPs) for three - dimensional partial differential equations are studied. The situation with them is rather surprising and there is no general understanding even more than sixty years after their statement given by Murray Protter. These problems are multidimensional analogues of classical BVPs o...

In this paper we introduce a time-dependent SEIR-based model with vaccination. In the suggested model the host population is divided into seven compartments: susceptible, exposed, infectious, recovered, deceased, vaccinated susceptible individuals and individuals with vaccination-acquired immunity. The dynamics of the infection in these groups is m...

We study some properties of degenerated singular integral operators with variable coefficients and related canonical functions for associated conjugation problems. The integral operators are given on the positive semi-axis and degenerate at zero. The canonical functions have a singularity at zero, and the order of singularity depends on the approac...

Data from the World Health Organization indicate that Bulgaria has the second-highest COVID-19 mortality rate in the world and the lowest vaccination rate in the European Union. In this context, to find the crucial epidemiological parameters that characterize the ongoing pandemic in Bulgaria, we introduce an extended SEIRS model with time-dependent...

In this paper we discuss the topic of correct setting for the equation \((-\varDelta )^s u=f\), with \(0<s <1\). The definition of the fractional Laplacian on the whole space \(\mathbb R^n\), \(n=1,2,3\) is understood through the Fourier transform, see, e.g., Karniadakis et al. (arXiv, 2018). The real challenge however represents the case when this...

This work is the continuation of the article [7], in which we are interested in the mathematical study of a system of nonstationary Navier-Stokes equations coupled to the energy equation in a bounded domain of R³. The considered system could model the flow of a fluid in a solar collector. We establish the existence and uniqueness of the weak soluti...

We apply Pohozhaev type identity to establish nonexistence of nontrivial regular solution of a problem of mixed type. The result is obtained in both supercritical and critical case with brief discussion on the derivation of Sobolev embedding constant. Since boundary conditions are imposed on elliptic boundary and part of hyperbolic boundaries we ap...

Since the end of 2019, with the outbreak of the new virus COVID-19, the world changed entirely in many aspects, with the pandemia affecting the economies, healthcare systems and the global socium. As a result from this pandemic, scientists from many countries across the globe united in their efforts to study the virsus's behavior and are attempting...

In this paper we discuss the topic of correct setting for the equation $(-\Delta )^s u=f$, with $0<s <1$. The definition of the fractional Laplacian on the whole space $\mathbb R^n$, $n=1,2,3$ is understood through the Fourier transform, see, e.g., Lischke et.al. (J. Comp. Phys., 2020). The real challenge however represents the case when this equat...

In this paper we explore a time-depended SEIR model, in which the dynamics of the infection in four groups from a selected target group (population), divided according to the infection, are modeled by a system of nonlinear ordinary differential equations. Several basic parameters are involved in the model: coefficients of infection rate, incubation...

A brief review of known results, open problems and new contributions to the treatment of the nonexistence of nontrivial solutions to nonlinear boundary value problems (BVPs) with power type nonlinearity whose linear part is of mixed type is given. The role of Pohožaev type identities is discussed with this regard and the application of nonexistence...

In this work we are interested in the mathematical study of a system of stationary Navier-Stokes equations coupled with the energy equation in a bounded domain in ℝ³. The considered system could model the flow of a fluid in a solar collector. We establish the existence and uniqueness of the weak solution of the variational problem in an appropriate...

We study a boundary value problem with the oblique derivative on the semicircle and mixed conditions on the diameter for the Helmholtz equation in a semidisk, and also its relation to a 3D problem for the Laplace equation.

For nonlinear equations of Gellerstedt type, uniqueness of generalized solutions to the degenerate hyperbolic Cauchy-Goursat problem on characteristic triangles will be established. For homogeneous supercritical nonlinearities, the uniqueness of the trivial solution in the class of generalized solutions will be approved by combining suitable Pohoža...

Two boundary value problems in which one of the conditions is nonlocal and contains a real parameter are studied for an equation of mixed type in a half-strip. Sufficient conditions for the unique solvability of these problems are obtained under some restrictions on the parameter.

A brief survey of known results, open problems and new contributions to the understanding of the nonexistence of nontrivial solutions to nonlinear boundary value problems (BVPs) whose linear part is of mixed elliptic-hyperbolic type is given. Crucial issues discussed include: the role of so-called critical growth of the nonlinear terms in the equat...

We study a boundary value problem for (3 + 1)-D weakly hyperbolic equations of Keldysh type (problem PK). The Keldysh-type equations are known in some specific applications in plasma physics, optics, and analysis on projective spaces. Problem PK is not well-posed since it has infinite-dimensional cokernel. Actually, this problem is analogous to a s...

A (3 + 1)-dimensional boundary value problem for equations of Keldysh type (the second kind) is studied. Such problems for equations of Tricomi type (the first kind) or for the wave equation were formulated by M.H. Protter (1954) as multidimensional analogues of Darboux or Cauchy-Goursat plane problems. Now, it is well known that Protter problems a...

A (3 + 1)-dimensional boundary value problem for equations of Keldysh type (the second kind) is studied. Such problems for equations of Tricomi type (the first kind) or for the wave equation were formulated by M.H. Protter (1954) as multidimensional analogues of Darboux or Cauchy-Goursat plane problems. Now, it is well known that Protter problems a...

This paper deals with Protter problems for Keldysh type equations in ℝ4. Originally such type problems are formulated by M. Protter for equations of Tricomi type. Now it is well known that Protter problems for mixed type equations of the first kind are ill-posed and for smooth right-hand side functions they have singular generalized solutions. In t...

We consider a (2+1)-D boundary value problem for degenerate hyperbolic equation, which is closely connected with transonic fluid dynamics. This problem was introduced by Protter in 1954 as a multi-dimensional analogue of the Darboux problem in the plain, which is known to be well-posed. However the (2+1)-D problem is overdetermined with infinitely...

A three dimensional boundary value problem for equations of Keldysh type involving lower order terms is studied. This problem is not correctly set, since it has an infinite-dimensional co-kernel. In order to avoid the infinite number of necessary conditions for classical solvability a notion for generalized solution is given. For small power of deg...

Using a global existence result due to A. Granas et al. [7] and the barrier strips technique, we study the solvability of the boundary value problem x ′′′ = f (t, x, x ′, x ′′), x(0) = A, x ′(0) = B, x ′′(1) = C where f :[0,1]×ℝ3→ℝ is continuous. The obtained sufficient conditions guarantee the existence of at least one solution in C 3 [0,1] which...

Recent numerical solutions and shock tube experiments have shown the existence of a complex reflection pattern, known as Guderley Mach reflection, which provides a resolution of the von Neumann paradox of weak shock reflection. In this pattern, there is a sequence of tiny supersonic patches, reflected shocks and expansion waves behind the triple po...

We study four-dimensional boundary value problems for the nonhomogeneous wave equation, which are analogues of Darboux problems on the plane. They were formulated by M.H. Protter in connection with BVPs for mixed type equations that model transonic flow phenomena. It is known that the unique generalized solution of Protter&apos;s problem may have s...

Some three-dimensional boundary value problems for equations of Keldysh type are studied. Such type problems, but for equations of Tricomi type are stated by M. H. Protter [25] as 3-D analogues of Darboux or Cauchy-Goursat plane problems. It is well known that in contrast of well-posedness of 2D problems, the Protter problems are strongly ill-posed...

Using the Topological transversality theorem and the barrier strips technique, we study the solvability of the initial value problem x″ = f(t,x,x′),x(0) = A,x′(0)=B, where the scalar function f(t,x,p) may be unbounded as p→B. Obtained results guarantee the existence of at least one solution in C1[0,T]∪C2(0,T]. Under additional assumptions it is mon...

We consider a planar Darboux-Goursat problem with singular coefficients. It is known that its unique solution (which is not a solution in the classical sense) may have strong power type singularity isolated at one boundary point even for very smooth functions in the right-hand side of the equation. In the present work we derive an exact asymptotic...

For the four-dimensional nonhomogeneous wave equation boundary value problems that are multidimensional analogues of Darboux problems in the plane are studied. It is known that for smooth right-hand side functions the unique generalized solution may have a strong power-type singularity at only one point. This singularity is isolated at the vertex...

Using barrier strip arguments, we investigate the existence of C [ 0 , T ] ∩ C 2 ( 0 , T ] -solutions to the initial value problem x ″ = f ( t , x , x ′ ) , x ( 0 ) = A , lim t → 0 + x ′ ( t ) = B , which may be singular at x = A and x ′ = B . MSC:34B15, 34B16, 34B18.

For linear and semilinear equations of Tricomi type, existence, uniqueness and qualitative properties of weak solutions to the degenerate hyperbolic Goursat problem on characteristic triangles will be established. For the linear problem, a robust $L^2$-based theory will be developed, including well-posedness, elements of a spectral theory, partial...

Some three-dimensional boundary value problems for mixed type equations of second kind are studied. Such type problems, but for mixed type equations of first kind are stated by M. Protter in the fifties. For hyperbolic-elliptic equations they are multidimensional analogue of the classical two-dimensional Morawetz-Guderley transonic problem. For hyp...

We consider some boundary value problems for a weakly hyperbolic equation, which are three-dimensional analogues of the Darboux problems on the plain. These problems arise in transonic fluid dynamics and they are introduced by Protter in 1952. As distinct from the planar Darboux problems, the Protter problems are not well posed since the homogeneou...

We study four-dimensional boundary value problems for the nonhomogeneous wave equation, which are analogues of Darboux problems on the plane. It is known that the unique generalized solution may have a strong power-type singularity at only one point. This singularity is isolated at the vertex O of the boundary light characteristic cone and does not...

Four-dimensional boundary value problems, which were formulated by Protter for the nonhomogeneous wave equation, are studied. They can be considered as multidimensional versions of the Darboux problems in ℝ 2 . Protter’s problem is not well posed in the frame of classical solvability. On the other hand, it is known that the unique generalized solut...

We consider the steady transonic small disturbance equations on a domain and with data that lead to a solution that depends on a single variable. After writing down the solution, we show that it can also be found by using a hodograph transformation followed by a partial Fourier transform. This motivates considering perturbed problems that can be so...

In the fifties M. Protter stated new three-dimensional (3D) boundary value problems (BVP) for mixed type equations of first kind. For hyperbolic-elliptic equations they are multidimensional analogue of the classical two-dimensional (2D) Morawetz-Guderley transonic problem. Up to now, in this case, not a single example of nontrivial solution to the...

In 1952 M. H. Protter introduced some boundary value problems for weakly hyperbolic equations in a domain bounded by two characteristic surfaces and non-characteristic plane region. Such problems arise in fluid dynamics. They are multidimensional analogues of the Darboux problems on the plain. The Protter problems are not well possed since the homo...

About 50 years ago M.H. Protter introduced boundary value problems that are multidimensional analogues of the classical plane Morawetz problems for equations of mixed hyperbolic-elliptic type that model transonic fluid flows. Up to now there are no general existence results for the Protter-Morawetz multidimensional problems, and an understanding of...

We study three-dimensional boundary value problems for the nonhomogeneous wave equation, which are analogues of the Darboux problems in ℝ 2 . In contrast to the planar Darboux problem the three-dimensional version is not well posed, since its homogeneous adjoint problem has an infinite number of classical solutions. On the other hand, it is known...

For the (2+1)-D wave equation Protter formulated (1952) some boundary value problems which are three-dimensional analogues of the Dar-boux problems on the plane. Protter studied these problems in a 3-D domain, bounded by two characteristic cones and by a planar region. Now it is well known that, for an infinite number of smooth functions in the rig...

Using barrier strip arguments, we investigate the existence of C 2 [ 0 , 1 ] -solutions to the Neumann boundary value problem f ( t , x , x ′ , x ′ ′ ) = 0 , x ′ ( 0 ) = a , x ′ ( 1 ) = b . MSC: 34B15.

Some three-dimensional (3D) problems for mixed type equations of first and second kind are studied. For equation of Tricomi type, they are 3D analogs of the Darboux (or Cauchy-Goursat) plane problem. Such type problems for a class of hyperbolic and weakly hyperbolic equations as well as for some hyperbolic-elliptic equations are formulated by M. Pr...

For the wave equation we study boundary value problems, which are three-dimensional analogues of Darboux-problems on the plane. It is shown that for n in h there exists a right hand side smooth function from C*(Ω), for which the corresponding unique generalized solution belongs to C*(Ω\O), but it has a strong power-type singularity at the point O....

Using barrier strip type arguments we investigate the existence of solutions of the boundary value problem x"=f(t,x),t Î (0,1),x(0)=A,x¢(1)=0,{x''=f(t,x),\;t\in(0,1),\;x(0)=A,\;x'(1)=0,} where the scalar function f(t, x) may be singular at x=A. Mathematics Subject Classification (2000)34B15-34B16-34B18 KeywordsBoundary value problem-second order d...

Boundary value problems introduced by M. H. Protter for weakly hyperbolic equations are studied in (2+1)-D domain, bounded by two characteristic surfaces and non-characteristic plane region. The Protter problems are not well posed since the homogeneous adjoint problems have infinitely many nontrivial classical solutions. It is known that the unique...

Four-dimensional boundary value problems for the nonhomogeneous wave equation are studied, which are analogues of Darboux problems in the plane. In the frame of classical solvability the problem is not Fredholm, since it has infinite-dimensional cokernel. Alternatively, the notion of generalized solution was introduced. It is known that the general...

In this article we investigate the existence of positive and/or negative solutions of a classes of four-point boundary-value problems for fourth-order ordinary differential equations. The assumptions in this article are more relaxed than the known assumptions. Our technique relies on the continuum property (connectedness and compactness) of the sol...

In this work, we consider boundary-value problems of the form $$f(t, x, x', x'') = 0, 0 0$$ , where the scalar function f(t, x, p, q) may be singular at x = 0. As far as we know, the solvability of the singular boundary-value problems of this form has not been treated yet. Here we try to fill in this gap. Examples illustrating our main result ar...

We prove the nonexistence of nontrivial solutions for some linear classical planar problems studied by Tricomi, Frankl' and Guderlay-Morawetz, with additional nonlinearity having supercritical or critical growth. The results follow from integral identities of Pohožaev type, suitably calibrated to achieve an invariance with respect to anisotropic di...

Fifty years ago, starting with a planar problem formulated and studied by Cathleen Morawetz [ 12 ] with strong connection to transonic flow phenomena , M. Protter [ 17 ] gave a statement of its 3D generalization for elliptic-hyperbolic Gellerstedt equations. Later, it appeared that, when considered only in the hyperbolic-parabolic part Ω − of the o...

Fifty years ago, starting with a planar problem formulated and studied by CATHLEEN MORETZ [12] with strong connection to transonic flow phenomena, M. PROTTER [17] gave a statement of its 3D generalization for elliptic-hyperbolic Gellerstedt equations. Later, it appeared that, when considered only in the hyperbolic-parabolic part Ω of the original d...

For semilinear partial differential equations of mixed elliptic-hyperbolic type with various boundary conditions, the nonexistence of nontrivial solutions is shown for domains which are suitably star-shaped and for nonlinearities with supercritical growth in a suitable sense. The results follow from integral identities of Pohožaev type which are su...

Four-dimensional boundary value problems for the nonhomogeneous wave equation are studied, which are analogues of Darboux problems in the plane. The smoothness of the right-hand side function of the wave equation is decisive for the behavior of the solution of the boundary value problem. It is shown that for each n∈N there exists such a right-hand...

In 1952 M. Protter formulated some boundary value problems (BVP) for hyperbolic equations which are three-dimensional analogues of the Darboux problems (or Cauchy-Goursat problems) on the plane. As well he studied such problems for weakly hyperbolic equations in 3D domain Ωm, bounded by two characteristic surfaces ∑1m and ∑2m, and by a plane region...

A method to solve a singular boundary value problem for the differential equation f(t, x, x′, x″) = 0 using barrier strips, is discussed. The method was devoted to the solvability of various singular BVPs for ordinary differential equations, whose nonlinearity does not depend on the highest derivative. Various nonsingular BVPs for second-order diff...

For 3-D wave equation M. Protter formulated (1952) some boundary value problems (BVP) which axe three-dimensional analogues of the Darboux problems on the plane. Protter studied these problems in a 3-D domain Ω0, bounded by two characteristic cones ∑1 and ∑2,0, and by a plane region ∑0. Now, 50 years later, it is well known that, for an infinite nu...

In 1952, for the wave equation,Protter formulated some boundary value problems (BVPs), which are multidimensional analogues of Darboux problems on the plane. He studied these problems in a 3D domain Ω0, bounded by two characteristic cones Σ1 and Σ2,0 and a plane region Σ0. What is the situation around these BVPs now after 50 years? It i...

For the wave equation we study boundary value problems, which are four-dimensional analogues of Darboux problems on the plane. It is shown that for n in ℕ there exists a right hand side smooth function from C , for which the corresponding unique generalized solution has a strong power-type singularity at the point O. This singularity is isolated at...

In 1952, at a conference in New York, Protter formulated some boundary value problems for the wave equation, which are three-dimensional analogues of the Darboux problems (or Cauchy-Goursat problems) on the plane. Protter studied these problems in a 3-D domain $Omega_0$, bounded by two characteristic cones $Sigma_1$ and $Sigma_{2,0}$, and by a plan...

In this paper we study boundary-value problems for the wave equation, which are three-dimensional analogue of Darboux-problems (or of Cauchy-Goursat problems) on the plane. It is shown that for $n$ in $mathbb{N}$ there exists a right hand side smooth function from $C^n(ar{Omega}_{0})$, for which the corresponding unique generalized solution belongs...

The existence results obtained for the Dirichlet and mixed BVPs for the equation x = f (t, x, x) are extended to BVPs with full nonlinear conditions. The proofs are based on the theorem of Granas, Guenther and Lee, while barrier strips are used to obtain a priori bounds for solutions.

In this paper we investigate some boundary value problems for the wave equation, which are the three-dimensional analogues of the Darboux problems (or Cauchy-Goursat problems) on the plane. It is well known that for an infinite number of smooth functions in the right-hand side these problems do not have a classical solution. We define an appropriat...

In this paper we investigate some boundary value problems for degenerate hyperbolic equations in R3 which are the three-dimensional analogues of the Darboux-problems (or Cauchy-Goursat problems) in R2. It is well known that the Darboux-problems in the plane are well posed, while the same is not true for the corresponding problems in R3. It turns ou...

Certain boundary value problems for the wave equation are considered which are a multidimensional generalization of the Darboux problem on a plane. The value range of the corresponding operator has an infinite codimensionality in L2. To study these problems, new boundary value problems are introduced not for the wave operator but for a certain nonl...

. Some three-dimensional analogues of the plane Darboux problems for hyperbolic equations with degeneracy are investigated. In 1954, Protter initiated the study of such threedimensional problems, and it is now well known that for an infinite number of smooth right-hand sides these problems have solutions with a strong power-type singularity on the...

We investigate some boundary value problems for the wave equation. M. H. Protter (1954) formulated those problems as some three-dimensional analogues of the Darboux-problems (or Cauchy-Goursat problems) on the plane. It is well-known that for an infinite number of smooth functions in the right-hand side these problems do not have a classical soluti...

Some three-dimensional analogous of the plane Darboux-problems for hyperbolic equations with degeneracy are investigated. In 1954, Protter initiated the study of such three-dimensional problems, and it is now well known, that for an infinite number of smooth right-hand sides these problems have solutions with a strong power-type singularity on the...

We investigate some boundary value problems in ℝ 3 for the Tricomi equation of mixed type. M. H. Protter initiated the study of these problems, which are three-dimensional analogues of the plane problems, formulated by C. Morawetz, and investigated by C. Morawetz, P. Lax and R. Phillips. It is known that in the hyperbolic part of the domain, for an...