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This paper offers a number of examples showing that in the case of two independent variables the uniform ellipticity of a linear system of differential equations with partial derivatives of the second order, which fulfills condition (3), do not always cause the normal solvability of formulated exterior elliptic problems in the sense of Noether. Nevertheless, from the system of differential equations with partial derivatives of elliptic type it is possible to choose, under certain additional conditions, classes which are normally solvable in the sense of Noether. This paper also shows that for the so-called decomposed system of differential equations, with partial derivatives of an elliptic type in the case of exterior regions, the Noether theorems are valid. [ABSTRACT FROM AUTHOR]

This paper offers a number of examples showing that in the case of two independent variables the uniform ellipticity of a linear system of differential equations with partial derivatives of the second order, which fulfills condition (3), do not always cause the normal solvability of formulated exterior elliptic problems in the sense of Noether. Nevertheless, from the system of differential equations with partial derivatives of elliptic type it is possible to choose, under certain additional conditions, classes which are normally solvable in the sense of Noether. This paper also shows that for the so-called decomposed system of differential equations, with partial derivatives of an elliptic type in the case of exterior regions, the Noether theorems are valid. [ABSTRACT FROM AUTHOR]

In the following paper, we have shown the existence and localization of solutions for a system of n third order differential equations under Sturm-Liouville type boundary conditions. Such systems appear in many physical problems, one of which is the jerk equations to locate the trajectory of a material point in space. [ABSTRACT FROM AUTHOR]

This paper deals with a boundary-value problem for a coupled chemotaxis-Stokes system with nonlinear diffusion and a complicated nonlinear coupling term. The mathematical model considered herein appears as CNF { n t + u ⋅ ∇ n = Δ n m − ∇ ⋅ (n ∇ c) + ∇ ⋅ (n ∇ ϕ) , x ∈ Ω , t > 0 , c t + u ⋅ ∇ c = Δ c − n c , x ∈ Ω , t > 0 , u t + ∇ P = Δ u − n ∇ ϕ + n ∇ c , x ∈ Ω , t > 0 , ∇ ⋅ u = 0 , x ∈ Ω , t > 0 , where Ω ⊆ R 3 is a general bounded domain with smooth boundary and ϕ ∈ W 2 , ∞ (Ω) is a given gravitational potential function. Here, one of the novelties is that both the effect of gravity (potential force) on cells and the effect of the chemotactic force on fluid is considered, which leads to the stronger coupling than usual chemotaxis-fluid model studied in the most existing literatures. Based on a new energy-type argument combined with maximal Sobolev regularity theory, we conclude that if m > 7 6 , an associated initial-boundary value problem admits at least one globally weak solution which is uniformly bounded. This extends a recent result by Wang and Zhao (J. Differ. Equ. 269:148–179, 2020) which asserts global existence of weak solutions under the constraints m > 4 3 . [ABSTRACT FROM AUTHOR]

In this paper, we study the following boundary value problem involving the weak p -Laplacian. − M (∥ u ∥ E p p) Δ p u = h (x , u) in S ∖ S 0 ; u = 0 on S 0 , where S is the Sierpiński gasket in R 2 , S 0 is its boundary, M : R + → R is defined by M (t) = a t k + b , where a , b , k > 0 and h : S × R → R . We will show the existence of two nontrivial weak solutions to the above problem. [ABSTRACT FROM AUTHOR]

We investigate in the paper the initial boundary value problem of the two dimensional incompressible Navier-Stokes-Darcy system in a strip domain. It is shown that, without any small initial data assumption, the Navier-Stokes-Darcy equations have a unique global strong solution in the strip domain with a flat interface. The key is to establish the global-in-time regularity uniformly by pursuing the properties of Dirichlet-Neumann operator. [ABSTRACT FROM AUTHOR]

In this paper we deal with a one-dimensional free-boundary problem arising in the modeling of concrete carbonation process. More precisely, we investigate global existence, uniqueness and large-time behavior of weak solutions for the problem under consideration. We also obtain the existence of a weak solution when the measure of the initial domain vanishes. [ABSTRACT FROM AUTHOR]

In this paper we prove existence of global strong-weak two-dimensional solutions to the Navier-Stokes and heat equations coupled by the external force dependent on temperature and the heat dissipation, respectively. The existence is proved in a bounded domain with the Navier boundary conditions for velocity and the Dirichlet boundary condition for temperature. Next, we prove existence of 3d global strong solutions via stability. [ABSTRACT FROM AUTHOR]

In this paper, we study the compressible Euler-Korteweg equations with free boundary condition in vacuum. Under physically assumptions of positive density and pressure, we introduce some physically quantities to show that the spreading diameter of regions grows linearly in time. This is an interesting result as one would expect that the capillary forces would prevent the boundary from spreading. Moreover, we construct a spherically symmetric global solution to support our theorem, followed by Sideris (J. Differ. Equ. 257:1-14, 2014). [ABSTRACT FROM AUTHOR]

In this paper, we study the regularity of 3d axisymmetric Navier-Stokes equations under a prior point assumption on $v^{r}$ or $v^{z}$ . That is, the weak solution of the 3d axisymmetric Navier-Stokes equations $v$ is smooth if where $r$ is the distance from the point $x$ to the symmetric axis. [ABSTRACT FROM AUTHOR]

In this paper we consider the system of reaction-diffusion-advection equations with a free boundary, which arises in a competition ecological model in heterogeneous environment. In strong competition case, we study the influence of competition rates on the long time behavior of solutions and prove that two species spatially segregate as the competition rates become large. Besides, by using a blow up method, we obtain the uniform Hölder bounds for solutions of the system. [ABSTRACT FROM AUTHOR]

In this paper, we derive explicit sharp two-sided estimates for the Dirichlet heat kernels of a large class of symmetric (but not necessarily rotationally symmetric) Lévy processes on half spaces for all $t>0$ . These Lévy processes may or may not have Gaussian component. When Lévy density is comparable to a decreasing function with damping exponent $\beta$ , our estimate is explicit in terms of the distance to the boundary, the Lévy exponent and the damping exponent $\beta$ of Lévy density. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the initial value problem for a class of semilinear impulsive stochastic parabolic functional differential equations. Incorporating certain positive operators, these equations have as archetype impulsive stochastic functional heat differential equations supplemented by homogeneous Dirichlet boundary conditions. By the classical Galerkin's method, we prove a well-posedness result for the initial value problem for a class of linear random evolution equation which is necessary in developing our main theory. Using this well-posedness result and the classical contraction mapping argument, we prove that the initial value problem under consideration is globally well-posed. Employing the technique of Razumikhin, we prove under some additional assumptions that the trivial solution of the equation in question is mean-square exponentially stable. [ABSTRACT FROM AUTHOR]

Enumeration problems for the central configurations of the Newtonian $n$ body problem are hard for $n>3$ in $\mathbb{R}^{2}$ and $n>4$ in $\mathbb{R}^{3}$. These are problems in finding the numbers of classes of central configurations for all the masses in a parameter space of positive dimensions. Many results are obtained generically. That is, rigorous proofs of the counting problems only exists for parameters not at the bifurcation points. For the bifurcation points, only numerical evidences are provided due to the complexity of the problems. In this paper, we propose an algorithm that rigorously proves results on counting central configurations for all masses in one dimensional parameter spaces. Especially, we provide an approach to find all bifurcation points and count real roots at those points, known only implicitly. A spatial restricted $(4+1)$-body problem and a planar $(1+3)$-body problem are successfully applied by our method. All results except for the equal masses for the restricted $(4+1)$-body problem are new and the results for the planar $(1+3)$-body problem are new at the bifurcation points. [ABSTRACT FROM AUTHOR]

In this paper we presents further developments regarding the enrichment of the basic Theory of Order Completion as presented in Oberguggenberger and Rosinger (Solution of continuous nonlinear PDEs through order completion, North-Holland, Amsterdam, ). In particular, spaces of generalized functions are constructed that contain generalized solutions to a large class of systems of continuous, nonlinear PDEs. In terms of the existence and uniqueness results previously obtained for such systems of equations (van der Walt, Acta Appl. Math. 103:1–17, ), one may interpret the existence of generalized solutions presented here as a regularity result. Furthermore, it is indicated how the methods developed in this paper may be adapted to solve initial and/or boundary value problems. In particular, we consider the Navier-Stokes equations in three spacial dimensions, subject to an initial condition on the velocity. In this regard, we obtain the existence of a generalized solution to a large class of such initial value problems. [ABSTRACT FROM AUTHOR]

In this paper we investigate a reaction-diffusion-advection equation with a free boundary and sign-changing coefficient. The main objective is to understand the influence of the advection term on the long time behavior of the solutions. More precisely, we prove a spreading-vanishing dichotomy result, namely the species either successfully spreads to infinity as $t\rightarrow\infty$ and survives in the new environment, or it fails to establish and dies out in the long run. When spreading occurs, the spreading speed of the expanding front is affected by the advection. In this situation, we obtain best possible upper and lower bounds for the spreading speed of the expanding front. Furthermore, when the environment is asymptotically homogeneous at infinity, these two bounds coincide. [ABSTRACT FROM AUTHOR]

This paper concerns with the stabilization of a Timoshenko beam with bounded constraints on boundary feedback controls. Since the resulting controlled system is nonlinear, the weak well-posedness is proven by theories of the nonlinear monotone operators and the optimization. Then, the asymptotical stability of the controlled beam is analyzed by the weak topology, and its exponential stability is also proven by the Lyapunov's second method. In the end, the numerical experiment indicates that the control design is feasible. [ABSTRACT FROM AUTHOR]

This paper is concerned with two examples on the application of the free boundary formulation to BVPs on a semi-infinite interval. In both cases we are able to provide the exact solution of both the BVP and its free boundary formulation. Therefore, these problems can be used as benchmarks for the numerical methods applied to BVPs on a semi-infinite interval and to free BVPs. Moreover, we emphasize how for two classes of free BVPs, we can define non-iterative initial value methods, whereas BVPs are usually solved iteratively. These non-iterative methods can be deduced within Lie's group invariance theory. Then, we show how to apply the non-iterative methods to the two introduced free boundary formulations in order to obtain meaningful numerical results. Finally, we indicate several problems from the literature where our non-iterative transformation methods can be applied. [ABSTRACT FROM AUTHOR]

In this paper, we study a compressible Primitive Equations (CPEs) of the ocean in the two dimensional space, with horizontal periodic and vertical mixed boundary conditions. Thanks to an effective change of variables, we obtain a new CPEs model, which is similar as viscous shallow water equation. Using a new entropy estimate, we prove the stability of weak solutions for this new two dimensional CPEs model. [ABSTRACT FROM AUTHOR]

We establish the global-in-time existence and uniqueness of classical solutions of the initial boundary value problem for 1D compressible magnetohydrodynamics with zero magnetic resistivity, where the initial vacuum is allowed. In present paper, we do not require initial value to be small. [ABSTRACT FROM AUTHOR]

In this paper, we consider the initial boundary value problem of a class of non-Newtonian fluids. We obtain that finite velocity of the propagation of perturbations. [ABSTRACT FROM AUTHOR]

The aim of this paper is to tackle the self-propelling at low Reynolds number by using tools coming from control theory. More precisely we first address the controllability problem: 'Given two arbitrary positions, does it exist 'controls' such that the body can swim from one position to another, with null initial and final deformations?'. We consider a spherical object surrounded by a viscous incompressible fluid filling the remaining part of the three dimensional space. The object is undergoing radial and axi-symmetric deformations in order to propel itself in the fluid. Since we assume that the motion takes place at low Reynolds number, the fluid is governed by the Stokes equations. In this case, the governing equations can be reduced to a finite dimensional control system. By combining perturbation arguments and Lie brackets computations, we establish the controllability property. Finally we study the time optimal control problem for a simplified system. We derive the necessary optimality conditions by using the Pontryagin maximum principle. In several particular cases we are able to compute the explicit form of the time optimal control and to investigate the variation of optimal solutions with respect to the number of inputs. [ABSTRACT FROM AUTHOR]

This paper concerns with the dynamical theory of elasticity for solids with double porosity. This theory unifies the earlier proposed quasi-static model of Aifantis of consolidation with double porosity. The basic properties of plane waves are established. The radiation conditions of regular vectors are given. The basic internal and external boundary value problems (BVPs) of steady vibrations are formulated. The uniqueness theorems are proved. The basic properties of elastopotentials are given. The existence of regular (classical) solution of the external BVP by means of the potential method (boundary integral method) and the theory of singular integral equations are proved. [ABSTRACT FROM AUTHOR]

In this paper, we establish the global existence and stability of a steady symmetric shock wave for the constant supersonic flow past an infinitely long and large curved conic body. The flow is assumed to be polytropic, isentropic and described by a steady potential equation. Through looking for the suitable 'dissipative' boundary conditions on the shock and the conic surface together with the special form of shock equation, we show that the conic shock attached at the vertex of the cone exists globally in the whole space when the speed of the supersonic incoming flow is appropriately large. [ABSTRACT FROM AUTHOR]

This paper deals with a diffusive logistic equation on one dimensional isotropically growing domain. The model equation on growing domains is first presented, and the comparison principle is then proved. The asymptotic behavior of temporal solutions to the reaction-diffusion problem is given by constructing upper and lower solutions. Our result shows that when the domain grows slowly, the species successfully spreads to the whole habitat and stabilizes at a positive steady state, while it dies out in the long run if the domain grows fast. Numerical simulations are also presented to illustrate the analytical result. [ABSTRACT FROM AUTHOR]

Our motivation is a mathematical model describing the spatial propagation of an epidemic disease through a population. In this model, the pathogen diversity is structured into two clusters and then the population is divided into eight classes which permits to distinguish between the infected/uninfected population with respect to clusters. In this paper, we prove the weak and the global existence results of the solutions for the considered reaction-diffusion system with Neumann boundary. Next, mathematical Turing formulation and numerical simulations are introduced to show the pattern formation for such systems. [ABSTRACT FROM AUTHOR]

This paper is concerned with a two-species predator-prey reaction-diffusion system with Beddington-DeAngelis functional response and subject to homogeneous Neumann boundary conditions. By linearizing the system at the positive constant steady-state solution and analyzing the associated characteristic equation in detail, the asymptotic stability of the positive constant steady-state solution and the existence of local Hopf bifurcations are investigated. Also, it is shown that the appearance of the diffusion and homogeneous Neumann boundary conditions can lead to the appearance of codimension two Bagdanov-Takens bifurcation. Moreover, by applying the normal form theory and the center manifold reduction for partial differential equations (PDEs), the explicit algorithm determining the direction of Hopf bifurcations and the stability of bifurcating periodic solutions is given. Finally, numerical simulations supporting the theoretical analysis are also included. [ABSTRACT FROM AUTHOR]

The Jimbo-Miwa equation is the second equation in the well known KP hierarchy of integrable systems, which is used to describe certain interesting (3+1)-dimensional waves in physics but not pass any of the conventional integrability tests. The Konopelchenko-Dubrovsky equations arose in physics in connection with the nonlinear weaves with a weak dispersion. In this paper, we obtain two families of explicit exact solutions with multiple parameter functions for these equations by using Xu's stable-range method and our logarithmic generalization of the stable-range method. These parameter functions make our solutions more applicable to related practical models and boundary value problems. [ABSTRACT FROM AUTHOR]

In this paper, we develop a numerical technique for singularly perturbed boundary value problems using B-spline functions and least square method. The approximate solution derived in this article is convergent to the exact solution and can be applied both to linear and nonlinear models. The numerical examples and computational results illustrate and guarantee a higher accuracy for this technique. [ABSTRACT FROM AUTHOR]

In this paper, the exact number of solutions and positive solutions for a second-order Neumann boundary value problem is considered. The exact multiplicity results for this problem are established by Sobolev inequality, comparison theorem, maximum principle and lower and upper solutions method. [ABSTRACT FROM AUTHOR]

In this paper we consider a transmission problem with a boundary damping condition of memory type, that is, the wave propagation over bodies consisting of two physically different types of materials. One component is clamped, while the other is in a viscoelastic fluid producing a dissipative mechanism on the boundary. We will study the global existence of solutions for the transmission problem, and moreover we show that if the relaxation function decays exponentially or polynomially, then the solutions for the problem have the same decay rates. [ABSTRACT FROM AUTHOR]

This paper studies the existence of a positive solution to the second-order periodic boundary value problem where the nonlinear term f( t, u) may be singular at t=0, t=2 π and u=0. When there exist the limit functions lim u→+0 f( t, u)/ u and lim u→+∞ f( t, u)/ u, we prove that the problem has a positive solution provided that the integrations of the limit functions are appropriate. [ABSTRACT FROM AUTHOR]

In this paper, the Mönch fixed point theorem is used to investigate the existence of solutions of a n-point boundary value problem on the half-line in a Banach space. As an application, we give an example in an infinite dimensional space to demonstrate our results. [ABSTRACT FROM AUTHOR]

In this paper, we consider exact controllability to trajectories for a semilinear heat equation with nonlinearity that has superlinear growth at infinity with Dirichlet boundary conditions in a bounded domain of ℝ N. The results are established by using the variational methods, the related duality theory and Kakutani Fixed-point Theorem and generalize the previous results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the spectral analysis of transport operator with general boundary conditions in L1-setting. This problem will be investigated under results from the theory of positive linear operators, irreducibility and regularity of the collision operator. The basic problems treated here are notions of essential spectra, spectral bound and leading eigenvalues. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of three positive solutions for the following impulsive boundary value problem where φ: R→ R is the increasing homeomorphism (see Definition 1.1) and positive homomorphism and φ(0)=0. Using the five functionals fixed point theorem, we provide sufficient conditions for the existence of three positive solutions for the above boundary value problems. [ABSTRACT FROM AUTHOR]

In this survey paper, we shall establish sufficient conditions for the existence and uniqueness of solutions for various classes of initial and boundary value problem for fractional differential equations and inclusions involving the Caputo fractional derivative. The both cases of convex and nonconvex valued right hand side are considered. The topological structure of the set of solutions is also considered. [ABSTRACT FROM AUTHOR]

In this paper, we consider the fractional boundary value problem where D is the standard Riemann-Liouville fractional derivative. By means of fixed point theorems, sufficient conditions are obtained that guarantee the existence of solutions to the above boundary value problem. The fractional modeling is a generalization of the classical integer-order differential equations and it is a very important tool for modeling the anomalous dynamics of numerous processes involving complex systems found in many diverse fields of science and engineering. [ABSTRACT FROM AUTHOR]

In this paper, we consider an elastic beam equation where the nonlinear term is a Carathéodory function and the boundary condition is nonhomogeneous. We construct an iterative sequence by the help of monotonic technique and prove that the sequence approximates successively to the solution of the equation under suitable assumptions. [ABSTRACT FROM AUTHOR]

The study of dynamic equations on time scales is an area of mathematics. It has been created in order to unify the study of differential and difference equations. In this paper, we consider the time-scale boundary value problems where $\mathbb{T}$ is a time scale. By means of Leggett-Williams fixed point theorem, sufficient conditions are obtained that guarantee the existence of at least three positive solutions to the above boundary value problem. The results obtained are even new for the special cases of difference dynamic equations (when $\mathbb{T}=\mathbb{Z}$ ) and differential dynamic equations (when $\mathbb{T}=\mathbb{R}$ ), as well as in the general time scale setting. [ABSTRACT FROM AUTHOR]

The paper examines the issue of existence of solutions to the steady Navier-Stokes equations in an exterior domain in ℝ2. The system is studied with nonhomogeneous slip boundary conditions. The main results proves the existence of weak solutions for arbitrary data. [ABSTRACT FROM AUTHOR]

Abstract  In this paper we consider a p-Laplacian equation with the homogeneous Dirichlet boundary condition. We establish a blow-up result for certain solution with positive initial energy. [ABSTRACT FROM AUTHOR]

Abstract   This paper is concerned with the spectral analysis of a one-velocity transport operator with Maxwell boundary condition in L 1-space. After a detailed spectral analysis it is shown that the associated Cauchy problem is governed by a C 0-semigroup. Next, we discuss the irreducibility of the transport semigroup. In particular, we show that the transport semigroup is irreducible. Finally, a spectral decomposition of the solutions into an asymptotic term and a transient one which will be estimated for smooth initial data is given. [ABSTRACT FROM AUTHOR]