Existence of Entropy Solution for Degenerate Parabolic-Hyperbolic Problem Involving p(x)-Laplacian with Neumann Boundary Condition

We consider a strongly non-linear degenerate parabolic-hyperbolic problem with p(x)-Laplacian diffusion flux function. We propose an entropy formulation and prove the existence of an entropy solution.

THE EXISTENCE AND LOCAL UNIQUENESS OF MULTI-PEAK SOLUTIONS TO A CLASS OF KIRCHHOFF TYPE EQUATIONS

In this paper,we study the existence and local uniqueness of multi-peak solutions to the Kirchhoff type equations-(ε^(2)a+εb∫_(R^(3))|■u|^(2))△u+V(x)u=u^(p),u>0 in R^(3),which concentrate at non-degenerate critical points of the potential function V(x),where a,b>0,1<p<5 are constants,andε>0 is a parameter.Applying the Lyapunov-Schmidt reduction method and a local Pohozaev type identity,we establish the existence and local uniqueness results of multi-peak solutions,which concentrate at{a_(i)}1≤i≤k,where{a_(i)}1≤i≤k are non-degenerate critical points of V(x)asε→0.

PROPERTIES OF SOLUTIONS TO A HARMONIC-MAPPING TYPE EQUATION WITH A DIRICHLET BOUNDARY CONDITION

In the present paper, we consider the problem {-△u=u^(β_(1))|■u|^(β_(2)),in Ω,u=0,on ■Ω,u>0,in Ω,(0.1) where β_(1), β_(2) > 0 and β_(1) + β_(2) < 1, and Ω is a convex domain in R~n. The existence, uniqueness,regularity and (2-β_(2))/(1-β_(1)-β_(2))-concavity of the positive solutions of the problem(0.1) are proven.

GLOBAL SOLUTIONS TO A HYDRODYNAMIC MODEL FOR SEMICONDUCTORS WITH VELOCITY RELAXATION

In this paper,we apply the method given in the paper“Zero relaxation time limits to a hydrodynamic model of two carrier types for semiconductors”(Mathematische Annalen,2022,382:1031–1046)to study the Cauchy problem for a one dimensional inhomogeneous hydrodynamic model of two-carrier types for semiconductors with the velocity relaxation.

The Existence of Meromorphic Solutions to Non-Linear Delay Differential Equations

In this paper, we study the existence of the transcendental meromorphic solution of the delay differential equations , where a(z) is a rational function, and are polynomials in w(z) with rational coefficients, k is a positive integer. Under the assumption when above equations own transcendental meromorphic solutions with minimal hyper-type, we derive the concrete conditions on the degree of the right side of them. Specially, when w(z)=0 is a root of , its multiplicity is at most k. Some examples are given here to illustrate that our results are accurate.

Periodic Solutions of Mixed Type p-Laplacian Equations with Deviating Arguments

Based on Mansevich-Mawhin continuation theorem and some analysis skill,some sufficient conditions for the existence of periodic solutions for mixed type p-Laplacian equation with deviating arguments are established,which are complement of previously known results.

MULTIPLE POSITIVE SOLUTIONS FOR QUASILINEAR ELLIPTIC PROBLEMS INVOLVING CONCAVE-CONVEX NONLINEARITIES AND MULTIPLE HARDY-TYPE TERMS

In this paper, we deal with the existence and multiplicity of positive solutions for the quasilinear elliptic problem -△pu-∑i=1^kμi｜u｜^p-2/｜x-ai｜p^u=｜u｜^p^＊-2u＋λ｜u｜^q-2u,x∈Ω,where Ω belong to R^N（N ≥ 3） is a smooth bounded domain such that the different points ai∈Ω,i= 1,2,...,k,0≤μi〈μ^-=（N-p/p）^p,λ〉0,1≤q〈p,and p^＊=p^N/N-p.The results depend crucially cn the parameters λ,q and μi for i=1,2,...,k.

The Existence of Periodic and Almost Periodic Solutions of Logistic Type Equation

In this paper, we study the Logistic type equation x= a（t）x -b（t）x^2＋ e（t）. Under the assumptions that e（t） is small enough and a（t）, b（t） are contained in some positive intervals, we prove that this equation has a positive bounded solution which is stable. Moreover, this solution is a periodic solution if a（t）, b（t） and e（t） are periodic functions, and this solution is an almost periodic solution if a（t）, b（t） and e（t） are almost periodic functions.

On Local Existence and Blow-Up of Solutions for Nonlinear Wave Equations of Higher-Order Kirchhoff Type with Strong Dissipation

In this paper, we study on the initial-boundary value problem for nonlinear wave equations of higher-order Kirchhoff type with Strong Dissipation: . At first, we prove the existence and uniqueness of the local solution by the Banach contraction mapping principle. Then, by “Concavity” method we establish three blow-up results for certain solutions in the case 1): , in the case 2): and in the case 3): . At last, we consider that the estimation of the upper bounds of the blow-up time is given for deferent initial energy.

Nonequivalent Similarity Reductions and Exact Solutions for Coupled Burgers-Type Equations

Using the machinery of Lie group analysis,the nonlinear system of coupled Burgers-type equations is studied.Using the infinitesimal generators in the optimal system of subalgebra of the said Lie algebras,it leads to two nonequivalent similarity transformations by using it we obtain two reductions in the form of system of nonlinear ordinary differential equations.The search for solutions of these systems by using the G'/G-method has yielded certain exact solutions expressed by rational functions,hyperbolic functions,and trigonometric functions.Some figures are given to show the properties of the solutions.

THE EXISTENCE OF PERIODIC SOLUTIONS OF IMPULSIVE DIFFERENTIAL EQUATIONS OF MIXED TYPE

The existence of periodic solutions for a class of impulsive differential equations of mixed type is studied by constructing periodic sequence solutions of difference equations.

A Chain of Type Ⅱ and Its Exact Solutions

The exact solutions of a chain of type Ⅱ are investigated. The chain of type Ⅱ is first transformed to an integrable differential-difference equation, which has the Kaup Newell spectral problem as its continuous spatial spectral problem and a Darboux transformation of the Kaup-Newell equation as its discrete temporal spectral problem. Then, with these spectral problems, a Darboux transformation of the transformed equation is constructed. Finally, as an application of the Darboux transformation, an exact solution of the transformed equation and thus the chain of type Ⅱ are presented.

Existence of Periodic Solutions for Neutral-Type Neural Networks with Delays on Time Scales

In this paper, we employ a fixed point theorem due to Krasnosel’skii to attain the existence of periodic solutions for neutral-type neural networks with delays on a periodic time scale. Some new sufficient conditions are established to show that there exists a unique periodic solution by the contraction mapping principle.

Global Existence and Large Time Asymptotic Behavior of Strong Solution to the Cauchy Problem of 2D Density-Dependent Boussinesq Equations of Korteweg Type

In this paper, we study the Cauchy problem of the density-dependent Boussinesq equations of Korteweg type on the whole space with a vacuum. It is proved that there exists a unique strong solution for the two-dimensional Cauchy problem established that the initial density and the initial temperature decay not extremely slow. Particularly, it is allowed to be arbitrarily large for the initial data and vacuum states for the initial density, even including the compact support. Moreover, when the density depends on the Korteweg term with the viscosity coefficient and capillary coefficient, we obtain a consistent priority estimate by the energy method, and extend the local strong solutions to the global strong solutions. Finally, when the pressure and external force are not affected, we deform the fluid models of Korteweg type, we can obtain the large time decay rates of the gradients of velocity, temperature and pressure.

Existence of Infinitely Many High Energy Solutions for a Fourth-Order Kirchhoff Type Elliptic Equation in R³

In this paper, we consider the following fourth-order equation of Kirchhoff type where a, b > 0 are constants, 3 < p < 5, V ∈ C (R3, R);Δ2: = Δ (Δ) is the biharmonic operator. By using Symmetric Mountain Pass Theorem and variational methods, we prove that the above equation admits infinitely many high energy solutions under some sufficient assumptions on V (x). We make some assumptions on the potential V (x) to solve the difficulty of lack of compactness of the Sobolev embedding. Our results improve some related results in the literature.

The consistent Riccati expansion and new interaction solution for a Boussinesq-type coupled system

Starting from the Davey-Stewartson equation, a Boussinesq-type coupled equation system is obtained by using a variable separation approach. For the Boussinesq-type coupled equation system, its consistent Riccati expansion （CRE） solvability is studied with the help of a Riccati equation. It is significant that the soliton--cnoidal wave interaction solution, expressed explicitly by Jacobi elliptic functions and the third type of incomplete elliptic integral, of the system is also given.

SIMILARITY TRANSFORMATION, THE STRUCTURE OF THE TRAVELING WAVES SOLUTION AND THE EXISTENCE OF A GLOBAL SMOOTH SOLUTION TO GENERALIZED KURAMOTO SIVASHINSKY TYPE EQUATIONS

The existence of a global smooth solution for the initial value problem of generalized Kuramoto-Sivashinsky type equations have been obtained. Similarty siolutions and the structure of the traveling waves solution for the generalized KS equations are discussed and analysed by using the qualitative theory of ODE and Lie's infinitesimal transformation respectively.

PREDICTION OF THE RHEOLOGICAL PROPERTIES OF STAR—TYPE BRANCHED POLYMER SOLUTIONS BY MEANS OF THE FIXED SPECTRUM MODEL

The Rouse-Zimm model with slippage was improved and the basic parameters of modelwere modified to explain the rheological properties of star-type branched polymersolutions. The theoretical results show good agreement with experimental data.