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In this article, we study a class of elliptic problems involving both local and nonlocal operators with different orders and a singular nonlinearity in combination with critical Hartree type nonlinearity (see problem $ (P_\lambda) $ (P λ) below). Using variational methods together with the critical point theory of nonsmooth analysis, we show the existence, the regularity and the multiplicity of weak solutions with respect to the parameter λ. [ABSTRACT FROM AUTHOR]

In this work, we consider boundary value problems with characteristic, non-characteristic, and two mixed lines of type change for a fractionally loaded equation. The equation under consideration is a loaded parabolic-hyperbolic type with a fractional integral operator, where the hyperbolic part is a characteristic load. By assuming the load is characteristic under necessary conditions on the given functions, we prove the unique solvability of the resulting integral equations derived from the formulated problems. Consequently, the problems also have unique solutions. [ABSTRACT FROM AUTHOR]

Overdetermined boundary value problems and the minimal operators generated by them are extremely important in the description of regular boundary value problems for differential equations, and are also widely used in the study of local properties of solutions. In addition, for inverse problems of mathematical physics arising from applications, when determining unknown data, it is necessary to study problems with overdetermined boundary conditions, which is reflected in the study of problems, including for hyperbolic equations and systems, arising in physics, geophysics, seismic tomography, geoelectrics, electrodynamics, medicine, ecology, economics and many other practical areas. Thus, the study of overdetermined boundary value problems is of both theoretical and applied interest. In this paper, a criterion for the regular solvability of the overdetermined Cauchy problem for the Gellerstedt equation and the minimal differential operator generated by it in a hyperbolic domain is established, as which both the case of a characteristic triangle and the case of a more general domain with fairly general assumptions about the boundary of the domain are considered. Due to overdetermined boundary conditions, the problem under consideration will be ill-posed in the general case, therefore, for its regular solvability, additional conditions must be imposed on the initial data. In other words, we have considered the inverse problem: to determine what requirements the initial data of the problem, in particular the right part of the Gellerstedt equation, should meet, in question, so that the overdetermined Cauchy problem is regularly solvable. The proof is based on the Gellerstedt potential, the properties of solutions of the Goursat problem in the characteristic triangle, and the properties of special functions. [ABSTRACT FROM AUTHOR]

This work is devoted to the study of the Cauchy problem for a loaded differential equation with the Bessel operator. When studying problems for loaded equations, the properties of Erdélyi-Kober operators are used as transformation operators concerning a relation. We obtain an explicit form of the solution to the Cauchy problem for a loaded one-dimensional differential equation. At the end of the work, we will show several examples on graphs. [ABSTRACT FROM AUTHOR]

Both smooth subsonic and transonic spiral flows to steady Euler–Poisson system with nonzero angular velocity and vorticity in a concentric cylinder are studied. On the one hand, we investigate the structural stability of smooth cylindrically symmetric subsonic flows under three-dimensional perturbations on the inner and outer cylinders. On the other hand, the structural stability of smooth transonic flows under the axi-symmetric perturbations is examined. There are no any restrictions on the background subsonic and transonic solutions. A deformation-curl-Poisson decomposition to the steady Euler–Poisson system is utilized to deal with the hyperbolic-elliptic mixed structure in the subsonic region. We emphasize that there is a special structure of the steady Euler–Poisson system which yields a priori estimates and uniqueness of the linearized elliptic system. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and regularity of a capacity solution for a coupled nonlinear anisotropic parabolic-elliptic system, where the elliptic component of the parabolic equation involves thermal conductivities $ a_{i}(u) $ a i (u) that satisfy $ \lim _{s\rightarrow +\infty }a_{i}(s)=0 $ lim s → + ∞ a i (s) = 0 for all $ i=1,\ldots,N $ i = 1 , ... , N. We work with anisotropic Sobolev spaces and use Schauder's fixed-point theorem to find weak solutions to approximate problems. In addition, we validate the convergence of a subsequence of approximate solutions to a capacity solution. [ABSTRACT FROM AUTHOR]

The generator of the semigroup associated with linear age-structured population models including spatial diffusion is shown to have compact resolvent. [ABSTRACT FROM AUTHOR]

We establish existence results for a class of mixed anisotropic and nonlocal p-Laplace equations with singular nonlinearities. We consider both constant and variable singular exponents. Our argument is based on an approximation method. To this end, we also discuss the necessary regularity properties of weak solutions of the associated non-singular problems. More precisely, we obtain local boundedness of subsolutions, the Harnack inequality for solutions and the weak Harnack inequality for supersolutions. [ABSTRACT FROM AUTHOR]

A model is developed to describe thermal convection in a mixture of fluids in a porous medium, where the layer is heated from below while simultaneously the fluid density at the base of the porous layer is greater than that higher up. In addition to buoyancy forces which are essentially due to gravity the fluid mixture is subject to Korteweg stresses which arise because of density gradients in the mixture. A complete stability analysis is provided and the critical Rayleigh number for convective motion is derived for both stationary and oscillatory convection and this is complemented with a global energy stability analysis. The analogous problem in a bidisperse porous medium is also briefly discussed. [ABSTRACT FROM AUTHOR]

In R 2 , a symmetric blunt body W b is fixed by smoothing out the tip of a symmetric wedge W 0 with the half-wedge angle θ w ∈ (0 , π 2) . We first show that if a horizontal supersonic flow of uniform state moves toward W 0 with a Mach number M ∞ > 1 being sufficiently large depending on θ w , then the half-wedge angle θ w is less than the detachment angle so that there exist two shock solutions, a weak shock solution and a strong shock solution, with the shocks being straight and attached to the vertex of the wedge W 0 . Such shock solutions are given by a shock polar analysis, and they satisfy entropy conditions. The main goal of this work is to construct a detached shock solution of the steady Euler system for inviscid compressible irrotational flow in R 2 ∖ W b . Especially, we seek a shock solution with the far-field state given as the strong shock solution obtained from the shock polar analysis. Furthermore, we prove that the detached shock forms a convex curve around the blunt body W b if the Mach number of the incoming supersonic flow is sufficiently large, and if the boundary of W b is convex. [ABSTRACT FROM AUTHOR]

For the equation of a mixed parabolic-hyperbolic type with a power degeneration on the type change line, the inverse problems to determine the time-dependent factors of right-hand sides are studied. Based on the formula for solving a direct problem, the solution of inverse problems is equivalently reduced to the solvability of loaded integral equations. Using the theory of integral equations, the corresponding theorems for the existence and uniqueness of the solutions of the stated inverse problems are proved and the explicit formulas for the solution have been given. [ABSTRACT FROM AUTHOR]

A compartment epidemic model for infectious disease spreading is investigated, where movement of individuals is governed by spatial diffusion. The model includes infection age of the infected individuals and assumes a logistic growth of the susceptibles. Global well-posedness of the equations within the class of nonnegative smooth solutions is shown. Moreover, spectral properties of the linearization around a steady state are derived. This yields the notion of linear stability which is used to determine stability properties of the disease-free and the endemic steady state. [ABSTRACT FROM AUTHOR]

In this article, we consider a combination of local and nonlocal Laplace equation with singular nonlinearities. For such mixed problems, we establish the existence of at least one weak solution for a parameter-dependent singular nonlinearity and existence of multiple solutions for perturbed singular nonlinearity. Our argument is based on the variational and approximation approach. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problems on the solvability and constructing solutions of one nonlocal boundary-value problem for the multidimensional, fourth-order, integro-differential Benney–Luke equation with degenerate kernel and spectral parameters. For various values of spectral parameters, necessary and sufficient conditions of the existence of a solution are obtained. The Fourier series for solutions of the problem corresponding to various sets of spectral parameters are obtained. For regular values of spectral parameters, the absolute and uniform convergence of the series and the possibility of their termwise differentiation with respect to all variables are proved. The problem is also examined studied for cases of irregular values of spectral parameters. [ABSTRACT FROM AUTHOR]

We investigate the distribution of electric potential in a circular cylinder containing a circular disk whose center is not on that of the cylinder. By satisfying the boundary conditions on the disk and the surface of the cylinder, an integral equation is obtained for the solution of the problem. It is shown that this integral equation reduces to the known Fredholm integral equation of the second kind when the disk is coaxial with the cylinder. [ABSTRACT FROM AUTHOR]

We consider the stability analysis of a two-dimensional model for post-burn contraction. The model is based on morphoelasticity for permanent deformations and combined with a chemical-biological model that incorporates cellular densities, collagen density, and the concentration of chemoattractants. We formulate stability conditions depending on the decay rate of signaling molecules for both the continuous partial differential equations-based problem and the (semi-)discrete representation. We analyze the difference and convergence between the resulting spatial eigenvalues from the continuous and semi-discrete problems. [ABSTRACT FROM AUTHOR]

We study the parabolic equations involving local and nonlocal p-Laplace-type operators. Under the assumption that 2 < p < ∞ , an intrinsic Harnack inequality of nonnegative weak solutions to such problem is established by employing the energy estimates, De Giorgi lemma, and measure theoretical argument. [ABSTRACT FROM AUTHOR]

We establish Harnack's estimates for positive weak solutions to a mixed local and nonlocal doubly nonlinear parabolic equation, of the type ∂ t | u | p - 2 u - div A x , t , u , D u + L u (x , t) = 0 , where the vector field A satisfies the p-ellipticity and growth conditions and the integro-differential operator L whose model is the fractional p-Laplacian. All results presented in this paper are provided by using sharp tools in the doubly nonlinear theory together with quantitative estimates. [ABSTRACT FROM AUTHOR]

In this paper, we study the large-time behavior of solutions to the Navier–Stokes/Allen–Cahn system of compressible viscous gases in half space. We prove that the corresponding stationary solutions of the small amplitude to the inflow problem for the Navier–Stokes/Allen–Cahn system are time-asymptotically stable. The proof of asymptotic stability is based on the basic energy method. [ABSTRACT FROM AUTHOR]

In this paper, we consider the stability of solutions to a class of cancer invasion model. This kind of model was first proposed by Chaplain and Lolas in 2005. In fact, they proposed two kinds of tumor invasion models in 2005 (Chaplain and Lolas in Math Models Methods Appl Sci 15(11):1685–1734, 2005) and 2006 (Chaplain and Lolas in Netw Heterog Media 1(3):399–439, 2006), respectively, which have similar structures, but their solutions have different properties. For convenience of distinction, we call them Model I and Model II, respectively. A common feature of the two models is that they both consider the remodeling of extracellular matrix. For the research on the stability of solutions of the two models, there are only the three papers (Hillen in Math Models Methods Appl Sci 23(01):165–198, 2013; Tao in SIAM J Math Anal 47(6): 4229–4250, 2015; Wang in J Differ Equ 260(9):6960–6988, 2016) on Model II by removing the remodeling of extracellular matrix, and the paper (Jin in Nonlinearity 33(10):5049–5079, 2020) on Model I. However, as far as we know, although the existence of solutions of this kind of model with porous medium diffusion has achieved fruitful results, there are no relevant results on stability even without considering the remodeling effect of extracellular matrix. In the present paper, we consider the stability of this kind of model with nonlinear diffusion, we find that the simultaneous emergence of haptotaxis, nonlinear diffusion and remodeling effect does bring essential difficulties to the study of stability, such as the method of constructing Lyapunov functional is no longer applicable. In this paper, we use some detailed analytical techniques to prove the global asymptotic stability of bounded solutions. [ABSTRACT FROM AUTHOR]

In this paper, we consider Carleman estimates and inverse problems for the coupled quantitative thermoacoustic equations. In part I, we establish Carleman estimates for the coupled quantitative thermoacoustic equations by assuming that the coefficients satisfy suitable conditions and taking the usual weight function φ ⁢ (x , t) = e λ ⁢ ψ ⁢ (x , t) , ψ ⁢ (x , t) = | x - x 0 | 2 - β ⁢ | t - t 0 | 2 + β ⁢ t 0 2 for x in a bounded domain in ℝ n with C 3 -boundary and t ∈ (0 , T) , where t 0 = T 2 . We will discuss applications of the Carleman estimates to some inverse problems for the coupled quantitative thermoacoustic equations in the succeeding part II paper [M. Cristofol, S. Li and Y. Shang, Carleman estimates and inverse problems for the coupled quantitative thermoacoustic equations. Part II: Inverse problems, preprint 2020, https://hal.archives-ouvertes.fr/hal-02863385]. [ABSTRACT FROM AUTHOR]

In present work studied the new boundary value problem for semi linear (Power-type nonlinearities) system equations of mixed hyperbolic -elliptic Keldysh type in the multivariate dimension with the changing time direction. Considered problem and equation belongs to the modern level partial differential equations. Applying methods of functional analysis, topological methods, "ε"-regularizing, and continuation by the parameter at the same time with aid of a prior estimates, under assumptions conditions on coefficients of equations of system, the existence and uniqueness of generalized and regular solutions of a boundary problem are established in a weighted Sobolev's space. In this work one of main idea consists of show that the new boundary value problem which investigated in case of linear system equations can be well-posed when added nonlinear terms to this linear system equations, moreover in this case constructed new weithged spaces, the identity between of strong and weak solutions is established. [ABSTRACT FROM AUTHOR]

Burn injuries can decrease the quality of life of a patient tremendously, because of esthetic reasons and because of contractions that result from them. In severe case, skin contraction takes place at such a large extent that joint mobility of a patient is significantly inhibited. In these cases, one refers to a contracture. In order to predict the evolution of post-wounding skin, several mathematical model frameworks have been set up. These frameworks are based on complicated systems of partial differential equations that need finite element-like discretizations for the approximation of the solution. Since these computational frameworks can be expensive in terms of computation time and resources, we study the applicability of neural networks to reproduce the finite element results. Our neural network is able to simulate the evolution of skin in terms of contraction for over one year. The simulations are based on 25 input parameters that are characteristic for the patient and the injury. One of such input parameters is the stiffness of the skin. The neural network results have yielded an average goodness of fit ( R 2 ) of 0.9928 (± 0.0013). Further, a tremendous speed-up of 19354X was obtained with the neural network. We illustrate the applicability by an online medical App that takes into account the age of the patient and the length of the burn. [ABSTRACT FROM AUTHOR]

This paper is concerned with the large time behavior of solutions to the Cauchy problem for a one-dimensional compressible non-isentropic Navier-Stokes/Allen-Cahn system which is a combination of the classical Navier-Stokes system with an Allen-Cahn phase field description. Motivated by the relationship between Navier-Stokes/Allen-Cahn and Navier-Stokes, the author can prove that the solutions to the one dimensional compressible non-isentropic Navier-Stokes/Allen-Cahn system tend time-asymptotically to the rarefaction wave, where the strength of the rarefaction wave is not required to be small. The proof is mainly based on a basic energy method. [ABSTRACT FROM AUTHOR]

The Jacobi elliptic function expansion scheme (JEFES) is broader than tan hyperbolic scheme, is derived to assemble new wave solutions for the modal of two long waves' interactions. It's proposing a new method to accumulate more general exact solutions (solitary and periodic) of nonlinear coupled Korteweg-de Vries (NCKDV) system and its craving to look up the works that are finished. It will be shown that limit on the solution may get more new solutions. [ABSTRACT FROM AUTHOR]

Coupled 1D nonlinear Schrödinger (CNLS) system is investigated with the help of exp (-φ(ξ)) expansion method for Soliton solutions. The obtained solutions will be articulated hyperbolic and transcendental form. The proposed algorithm explicitly reveals high appositeness and affluence by numerical solutions. A graphical presentation is also cited. [ABSTRACT FROM AUTHOR]

In this paper, we study a boundary-value problem with discontinuous conjugation conditions on the line of type changing for a model equation of mixed hyperbolic-parabolic type with degeneration of order in the hyperbolicity domain. In the parabolic domain, the equation is the fractional diffusion equation, whereas in the hyperbolic domain it is the loaded one-speed transfer equation. We prove the uniqueness and existence theorem and propose an explicit solution of the problem in the parabolic and hyperbolic domains. [ABSTRACT FROM AUTHOR]

In this paper, a theorem on a priori estimates of solutions of the Tricomi problem for a second-order, mixed-type equation with the Gellerstedt operator in the hyperbolicity domain is proved. The a priori estimate obtained implies the uniqueness of the regular solution of the problem considered. [ABSTRACT FROM AUTHOR]

In this work, we provide mathematical and numerical analyses for a thermoviscoelastic nonlinear beam model with Coulomb friction dry law. Since the dynamic frictional conditions are nonsmooth, a regularization technique with smoothing parameters is applied to approximate a nonlinear variational formulation. We prove the existence of weak solutions satisfying the regularized variational formulation based on a priori estimates and results for a pseudomonotone operator. Then, we pass to limits, as the smoothing parameters tend to be zero, in order to show convergence results for the regularized formulation. We propose the fully discrete numerical schemes in which a guarded Newton method is used to compute fully discrete numerical approximations of a nonlinear system at each time step. Numerical experiments are performed with selected data to present numerical simulations as well. [ABSTRACT FROM AUTHOR]

We consider a one-dimensional morphoelastic model describing post-burn scar contraction. Contraction can lead to a limited range of motion (contracture). Reported prevalence of burn scar contractures are 58.6% at 3–6 weeks and 20.9% at 12 months post-reconstructive surgery after burns. This model describes the displacement of the dermal layer of the skin and the development of the effective Eulerian strain in the tissue. Besides these components, the model also contains components that play a major role in the skin repair after trauma. These components are signaling molecules, fibroblasts, myofibroblasts, and collagen. We perform a sensitivity analysis for many parameters of the model and use the results for a feasibility study. In this study, we test whether the model is suitable for predicting the extent of contraction in different age groups. To this end, we conduct an extensive literature review to find parameter values. From the sensitivity analysis, we conclude that the most sensitive parameters are the equilibrium collagen concentration in the dermal layer, the apoptosis rate of fibroblasts and myofibroblasts, and the secretion rate of signaling molecules. Further, although we can use the model to simulate significant distinct contraction densities in different age groups, our results differ from what is seen in the clinic. This particularly concerns children and elderly patients. In children we see more intense contractures if the burn injury occurs near a joint, because the growth induces extra forces on the tissue. Elderly patients seem to suffer less from contractures, possibly because of excess skin. [ABSTRACT FROM AUTHOR]