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In a previous paper, we have described all reproductive general solutions of a Post equation, supposing that a general solution is known. In this paper we describe all general solutions of Post equation, supposing that a general solution of this equation is known (Theorem 6). As a special case we get the previous characterization of reproductive solutions and a similar result for Boolean equations (Theorem 9). [ABSTRACT FROM AUTHOR]

The backward heat equation is a typical ill-posed problem. In this paper, we shall apply a dual least squares method connecting Shannon wavelet to the following equation Motivated by Regińska's work, we shall give two nonlinear approximate methods to regularize the approximate solutions for high-dimensional backward heat equation, and prove that our methods are convergent. [ABSTRACT FROM AUTHOR]

We make a further advance concerning the maximum principle for second-order elliptic operators. We investigate in particular a geometric condition, first considered by Berestycki–Nirenberg– Varadhan, that seems to be natural in view of the application of the boundary weak Harnack inequality, on which our argument is based. Setting it free from some technical assumptions, apparently needed in earlier papers, we significantly enlarge the class of unbounded domains where the maximum principle holds, compatibly with the first-order term. [ABSTRACT FROM AUTHOR]

We will show in this paper that if λ is very close to 1, then can be attained, where M is a compact–manifold with boundary. This result gives a counter–example to the conjecture of de Figueiredo and Ruf in their paper titled "On an inequality by Trudinger and Moser and related elliptic equations" ( Comm. Pure. Appl. Math., 55, 135–152, 2002). [ABSTRACT FROM AUTHOR]

In this paper, we obtain the necessary and sufficient conditions on the global existence of all positive (weak) solutions to a nonlinear degenerate parabolic equation with nonlinear boundary condition. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of a uniform attractor for non-autonomous Brinkman-Forchheimer equations with general delay and time-dependent external force. [ABSTRACT FROM AUTHOR]

In this paper, we study the energy equality and the uniqueness of weak solutions to the MHD equations in the critical space L∞(0, T; L n(Ω)). We prove that if the velocity u belongs to the critical space L∞(0, T; L n(Ω)), the energy equality holds. On the basis of the energy equality, we further prove that the weak solution to the MHD equations is unique. [ABSTRACT FROM AUTHOR]

In this paper, it is proved that with at most $$ O{\left( {N^{{\frac{{65}} {{66}}}} } \right)} $$ exceptions, all even positive integers up to N are expressible in the form $$ p^{2}_{2} + p^{3}_{3} + p^{4}_{4} + p^{5}_{5} $$ . This improves a recent result $$ O{\left( {N^{{\frac{{19193}} {{19200}} + \varepsilon }} } \right)} $$ due to C. Bauer. [ABSTRACT FROM AUTHOR]

In this paper we study a problem concerning the compulsory behavior of the solutions of systems of discrete equations u( k + 1) = F( k, u( k)), k ∈ N( a) = { a, a + 1, a + 2, . . . }, a ∈ ℕ, ℕ = {0, 1, . . . } and F : N( a) × ℝ n → ℝ n . A general principle for the existence of at least one solution with graph staying for every k ∈ N( a) in a previously prescribed domain is formulated. Such solutions are defined by means of the corresponding initial data and their existence is proved by means of retract type approach. For the development of this approach a notion of egress type points lying on the defined boundary of a given domain and with respect to the system considered is utilized. Unlike previous investigations, the boundary can contain points which are not points of egress type, too. Examples are inserted to illustrate the obtained result. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to give an elementary method for the computation of the cohomology groups $$ H^{q} {\left( {X,\Omega ^{p}_{X} {\left( L \right)}} \right)} $$ , (0 ≤ q ≤ n) of an n–dimensional non–primary Hopf manifold X with arbitrary fundamental group. We use the method of Zhou to generalize the results for primary Hopf manifolds and non–primary Hopf manifold with an Abelian fundamental group. [ABSTRACT FROM AUTHOR]

In this paper, we study the regularity of solutions for two–dimensional Cahn–Hilliard equation with non–constant mobility. Basing on the L p type estimates and Schauder type estimates, we prove the global existence of classical solutions. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the interior regularity including the local boundedness and the interior Holder continuity of weak solutions for parabolic equations of the p(x, t)-Laplacian type. We improve the Moser iteration technique and generalize the known results for the elliptic problem to the corresponding parabolic problem. [ABSTRACT FROM AUTHOR]

In this paper, we determine the general solution of the functional equation f1(2 x + y) + f2(2 x − y) = f3( x + y) + f4( x − y) + f5( x) without assuming any regularity condition on the unknown functions f1, f2, f3, f4, f5 : ℝ → ℝ. The general solution of this equation is obtained by finding the general solution of the functional equations f(2 x + y) + f(2 x − y) = g( x + y) + g( x − y) + h( x) and f(2 x + y) + f(2 x − y) = g( x + y) + g( x − y). The method used for solving these functional equations is elementary but exploits an important result due to Hosszú. The solution of this functional equation can also be determined in certain type of groups using two important results due to Székelyhidi. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the initial value problems of the nonlinear Kaup–Kupershmidt equations $$ \frac{{\partial u}} {{\partial t}} + a_{1} \frac{{u\partial ^{2} u}} {{\partial x^{2} }} + \beta \frac{{\partial ^{3} u}} {{\partial x^{3} }} + \gamma \frac{{\partial ^{5} u}} {{\partial x^{5} }} = 0,$$ ( x, t) ∈ R2, and $$ \frac{{\partial u}} {{\partial t}} + a_{2} \frac{{\partial u}} {{\partial x}}\frac{{\partial ^{2} u}} {{\partial x^{2} }} + \beta \frac{{\partial ^{3} u}} {{\partial x^{3} }} + \gamma \frac{{\partial ^{5} u}} {{\partial x^{5} }} = 0, $$ ( x, t) ∈ R2. Several important Strichartz type estimates for the fundamental solution of the corresponding linear problem are established. Then we apply such estimates to prove the local and global existence of solutions for the initial value problems of the nonlinear Kaup–Kupershmidt equations. The results show that a local solution exists if the initial function u0( x) ∈ Hs ( R), and s ≥ 5/4 for the first equation and s ≥ 301/108 for the second equation. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the positive solutions of strongly coupled nonlinear parabolic systems with nonlinear boundary conditions: Under appropriate hypotheses on the functions a, b, g, h, d and f, we obtain that the solutions may exist globally or blow up in finite time by utilizing upper and lower solution techniques. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the growth and fixed points of solutions and their 1st, 2nd derivatives, differential polynomial of second order linear differential equations with meromorphic coefficients, and obtain that the exponents of convergence of these fixed points are all equal to the order of growth. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the weak solutions $$ u \in W^{{1,p}}_{{{\text{loc}}}} {\left( \Omega \right)}{\left( {1 < p < \infty } \right)} $$ of the following equation with vanishing mean oscillation coefficients A( x): belong to $$ W^{{1,q}}_{{{\text{loc}}}} {\left( \Omega \right)}{\left( {\forall q \in {\left( {p,\infty } \right)}} \right)} $$, provided $$ F \in L^{q}_{{{\text{loc}}}} {\left( \Omega \right)} $$ and B( x, u, h) satisfies proper growth conditions, where Ω ⊂ R N ( N ≥ 2) is a bounded open set, A( x) = ( A ij( x)) N× N is a symmetric matrix function. [ABSTRACT FROM AUTHOR]

In this paper, we consider a vibrating beam with one segment made of viscoelastic material of a Kelvin–Voigt (shorted as K–V) type and other parts made of elastic material by means of the Timoshenko model. We have deduced mathematical equations modelling its vibration and studied the stability of the semigroup associated with the equation system. We obtain the exponential stability under certain hypotheses of the smoothness and structural condition of the coefficients of the system, and obtain the strong asymptotic stability under weaker hypotheses of the coefficients. [ABSTRACT FROM AUTHOR]

By means of the continuation theorem of coincidence degree theory, some new results on the non-existence, existence and unique existence of periodic solutions for a kind of second order neutral functional differential equation are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we prove the generalized Hyers-Ulam-Rassias stability of universal Jensen’s equations in Banach modules over a unitalC*-algebra. It is applied to show the stability of universal Jensen’s equations in a Hilbert module over a unitalC*-algebra. Moreover, we prove the stability of linear operators in a Hilbert module over a unitalC*-algebra. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the regularity of solutions to the Dirichlet problems for general second-order elliptic equations in Lebesgue and Morrey spaces. We consider both nondivergence and divergence forms and the coefficients of principle terms are assumed to be in VMO. [ABSTRACT FROM AUTHOR]

In this paper, we study and discuss the existence of multiple solutions of a class of non-linear elliptic equations with Neumann boundary condition, and obtain at least seven non-trivial solutions in which two are positive, two are negative and three are sign-changing. The study of problem (1.1):is based on the variational methods and critical point theory. We form our conclusion by using the sub-sup solution method, Mountain Pass Theorem in order intervals, Leray-Schauder degree theory and the invariance of decreasing flow. [ABSTRACT FROM AUTHOR]

This is a set of lecture notes for the first author's lectures on the difference equations in 2019 at the Institute of Advanced Study for Mathematics at Zhejiang University. We focus on explicit computations and examples. The convergence of local solutions is discussed. [ABSTRACT FROM AUTHOR]

Let u be a solution to a second order elliptic equation with singular potentials belonging to Kato–Fefferman–Phong’s class in Lipschitz domains. An elementary proof of the doubling property for u 2 over balls is presented, if the balls are contained in the domain or centered at some points near an open subset of the boundary on which the solution u vanishes continuously. Moreover, we prove the inner unique continuation theorems and the boundary unique continuation theorems for the elliptic equations, and we derive the ℬ p weight properties for the solution u near the boundary. [ABSTRACT FROM AUTHOR]

We establish the existence of fundamental solutions for the anisotropic porous medium equation, $$ u_{t} = {\sum\nolimits_{i = 1}^n {{\left( {u^{{m_{i} }} } \right)}x_{i} x_{i} } } $$ in ℜ n × (0,∞), where m1, m2, . . . , and m n are positive constants satisfying $$ \min _{{1 \leqslant i \leqslant n}} {\left\{ {m_{i} } \right\}} \leqslant 1,{\sum\nolimits_{i = 1}^n {m_{i} > n - 2,} } $$ and $$ \max _{{1 \leqslant i \leqslant n}} {\left\{ {m_{i} } \right\}} \leqslant \frac{1} {n}{\left( {2 + {\sum\nolimits_{i = 1}^n {m_{i} } }} \right)}. $$ [ABSTRACT FROM AUTHOR]

We consider the system of four linear matrix equationsA1X=C1,XB2 =C2,A3XB3 =C3 andA4XB4 =C4 over R, an arbitrary von Neumann regular ring with identity. A necessary and sufficient condition for the existence and the expression of the general solution to the system are derived. As applications, necessary and sufficient conditions are given for the system of matrix equationsA1X=C1 andA3X=C3 to have a bisymmetric solution, the system of matrix equationsA1X=C1 andA3XB3 =C3 to have a perselfconjugate solution over R with an involution and char R ?2, respectively. The representations of such solutions are also presented. Moreover, some auxiliary results on other systems over R are obtained. The previous known results on some systems of matrix equations are special cases of the new results. [ABSTRACT FROM AUTHOR]

Assuming that the external forces of the system are small enough, the reference temperature being a periodic function, we study the existence, the uniqueness and the regularity of time-periodic solutions for the Boussinesq equations in several classes of unbounded domains of ℝ n. Our analysis is based on the framework of weak- L p spaces. [ABSTRACT FROM AUTHOR]

The author provides a finer local as well as semilocal convergence analysis of a certain class of Broyden-like methods for solving equations containing a nondifferentiable term on the m-dimensional Euclidean space ( m ≥ 1 a natural number). [ABSTRACT FROM AUTHOR]

Let R be an exchange ring with primitive factors artinian. We prove that there exists a u∈ U( R) such that 1 R ± u ∈ U( R), if and only if for any a∈ R, there exists a u ∈ U( R) such that a ± u ∈ U( R). Furthermore, we prove that, for any A ∈ M n ( R)( n ≥ 2), there exists a U ∈ GL n ( R) such that A ± U ∈ GL n ( R). [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem of a shallow water equation and its local wellposedness. [ABSTRACT FROM AUTHOR]

Let G be a portion of aC1,a boundary of ann-dimensional domainD. Let u be a solution to a second order parabolic equation inD× (-T, T) and assume thatu= 0 on G × (-T, T), 0 ? G. We prove that u satis.es a three cylinder inequality near G × (-T, T) . As a consequence of the previous result we prove that ifu(x, t) =O(|x|k) for everyt? (-T, T) and everyk? N, thenuis identically equal to zero. [ABSTRACT FROM AUTHOR]

We prove the existence of solutions of the static Landau-Lifshitz equation with multidirect effective field and with Dirichlet boundary condition, and establish the stability of the solution of Landau-Lifshitz equation with respect to time. [ABSTRACT FROM AUTHOR]

Interface problems for second order quasi-linear elliptic partial differential equations in a two-dimensional space are studied. We prove that each weak solution can be decomposed into two parts near singular points, one of which is a finite sum of functions of the formcra logmr? (?), where the coefficientscdepend on theH1-norm of the solution, theC0,d-norm of the solution, and the equation only; and the other one of which is a regular one, the norm of which is also estimated. [ABSTRACT FROM AUTHOR]

We establish a criterion for the uniqueness of periodic solutions for a class of second-order equations. We also give an application to a polynomial system and corrections to some known results. [ABSTRACT FROM AUTHOR]