Doubly Degenerate Parabolic Equation with Nonlinear Inner and Boundary Sources

This paper deals with blow-up criterion for a doubly degenerate parabolic equation of the form （u^n）t = （｜ux｜^m-1ux）x ＋ u^p in （0, 1） × （0,T） subject to nonlinear boundary source （｜ux｜^m-1ux）（1,t） = u^q（1,t）, （｜ux｜^m-1ux）（0,t） = O, and positive initial data u（x,0） = uo（x）, where the parameters va, n, p, q 〉0. It is proved that the problem possesses global solutions if and only if p ≤ n and q〈min {n,m（n＋1）/m＋1}.

Blow Up for Solutions of Nonlinear Doubly Degenerate Parabolic Equation

This paper studies the conditions of blow up in finite time of solutions of initial boundary value problem for a class nonlinear doubly degenerate parabolic equation.

Hlder Continuity of the Gradient of Solutions of Nonlinear Degenerate Parabolic Systems

§1. Introduction In this paper we consider the parabolic system (?)t/(?)ui-div(|▽u|p-2▽ui)=0(1≤i≤m), with p>1, where ui=ui(x, t), ▽=gradxand x varies in an open domain Ω(?)RN. The system is degenerate if p>2 or singular if 1<p<2. A vector function u=(u1, u2, …, um) defined in ΩT=Ω×[0, T] is called a solution of the system (1.1) if

INITIAL AND BOUNDARY PROBLEMS FOR CERTAIN NONLINEAR DEGENERATR PARABOLIC SYSTEMS

In this paper we are concerned with the following nonlinear degenerate parabolic systems u_t=△x(gradψ(u))+D_xb(u)+f(x.t.u)with Dirichlet boundary conditions,where u,gradψ(u),b and f are vector valued functions and xUnder some structure conditions on the terms of the systems,we have established theresults on existence and uniquence of global solutions of the systems.

EXISTENCE AND UNIQUENESS FOR SECOND-ORDER VECTOR BOUNDARY VALUE PROBLEM OF NONLINEAR SYSTEMS

This paper is concerned with the following second-order vector boundary value problem ：x^R=f（t,Sx,x,x＇）,0〈t〈1,x（0）=A,g（x（1）,x＇（1））=B,where x,f,g,A and B are n-vectors. Under appropriate assumptions,existence and uniqueness of solutions are obtained by using upper and lower solutions method.

Existence of Non-trivial Nonnegative Periodic Solutions for a Nonlinear Diffusion System

In this paper, by the method of upper and lower solutions, we establish the existence of the non-trivial nonnegative periodic solutions for a class of degenerate diffusion system arising from dynamics of biological groups.

Entropy Formulation for Triply Nonlinear Degenerate Elliptic-Parabolic-Hyperbolic Equation with Zero-Flux Boundary Condition

In this note, we investigated existence and uniqueness of entropy solution for triply nonlinear degenerate parabolic problem with zero-flux boundary condition. Accordingly to the case of doubly nonlinear degenerate parabolic hyperbolic equation, we propose a generalization of entropy formulation and prove existence and uniqueness result without any structure condition.

Existence Results for Systems of Nonlinear Caputo Fractional Differential Equations

We aim, in this work, to demonstrate the existence of minimal and maximal coupled quasi-solutions for nonlinear Caputo fractional differential systems with order q ∈ (1,2). Our approach is based on mixed monotone iterative techniques developed under the concept of lower and upper quasi-solutions. Our results extend those obtained for ordinary differential equations and fractional ones.

EXISTENCE AND UNIQUENESS FOR THIRD ORDER NONLINEAR BOUNDARY VALUE PROBLEMS

In this paper, the existence and uniqueness of solutions for boundary valueproblem x′′′=f(t, x, x′, x″), x(0)=A, x′(0)=B, g(x′(1), x″(1))=0 are studied byusing Volterra type operator and upper and lower solutions. Our results improve someknown works.

ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO THE HEAT EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS

The purpose of this paper is to investigate the stability and asymptotic behavior of the time-dependent solutions to a linear parabolic equation with nonlinear boundary condition in relation to their corresponding steady state solutions. Then, the above results are extended to a semilinear parabolic equation with nonlinear boundary condition by analyzing the corresponding eigenvalue problem and using the method of upper and lower solutions.

Singular perturbation of a second-order three-point boundary value problem for nonlinear systems

This paper deals with the existence of solutions to a singularly perturbed second-order three-point boundary value problem for nonlinear differential systems. The authors construct an appropriate generalized lower- and upper-solution pair, a concept defined in this paper, and employ the Nagumo conditions and algebraic boundary layer functions to ensure the existence of solutions of the problem. The uniformly valid asymptotic estimate of the solutions is given as well. The differential systems have nonlinear dependence on all order derivatives of the unknown.

Nonexistence and Longtime Behaviors of Solutions to a Class of Nonlinear Degenerate Parabolic Equations Not in Divergence Form

In this paper, we study the nonexistence and longtime behavior of weak solution for the degenerate parabolic equation σtun=umdiv（｜▽um｜p-2▽um）＋γ｜▽um｜p＋βun with zero boundary condition. Blow-up time is derived when the blow-up does occur.

Traveling wave solutions in a nonlocal dispersal SIR epidemic model with nonlocal time-delay and general nonlinear incidences

This paper investigates a nonlocal dispersal epidemic model under the multiple nonlocal distributed delays and nonlinear incidence effects.First,the minimal wave speed c*and the basic reproduction number Ro are defined,which determine the existence of traveling wave solutions.Second,with the help of the upper and lower solutions,Schauder's fixed point theorem,and limiting techniques,the traveling waves satisfying some asymptotic boundary conditions are discussed.Specifically,when Ro>1,for every speed c>c^(*) there exists a traveling wave solution satisfying the boundary conditions,and there is no such traveling wave solution for any 0<c<c^(*) when R_(0)>1 or c>0 when R_(0)<1.Finally,we analyze the effects of nonlocal time delay on the minimum wave speed.

Nonlinear boundary value problems for discontinuous delayed differential equations

In this paper nonlinear boundary value problems for discontinuous delayed differen- tial equations are considered. Some existence and boundedness results of solutions are obtained via the method of upper and lower solutions, which may be discontinuous. Our analysis can be applied to those phenomena from physics and control theory which have been successfully described by delayed differential equations with discontinuous functions, such as electric, pneu- matic, and hydraulic networks. It is also an important step to precede the design of control signals when finite-time or practical stability control are concerned.

ASYMPTOTIC SOLUTION OF SINGULARLY PERTURBED PROBLEM FOR A NONLINEAR SINGULAR EQUATION

The singularly perturbed initial value problem for a nonlinear singular equation is considered. By using a simple and special method the asymptotic behavior of solution is studied.

ON THE EXISTENCE OF SOLUTION OF A NONLINEAR TWO-POINT BOUNDARY VALUE PROBLEM ARISING FROM A LIQUID METAL FLOW

In this paper, we discuss the existence of solution of a nonlinear two-point boundary value problem with a positive parameter Q arising in the study of surfacetension-induced flows of a liquid metal or semiconductor. By applying the Schauder's fixed-point theorem, we prove that the problem admits a solution for 0 ≤ Q ≤ 14.306.It improves the result of 0 ≤ Q ＜ 1 in [2] and 0 ≤ Q ≤ 13.213 in [3].

On the Cauchy problem for a doubly nonlinear degenerate parabolic equation with strongly nonlinear sources

In this article, we consider the existence of local and global solution to the Cauchy problem of a doubly nonlinear equation. By introducing the norms |||f||| h and 〈f〉h, we give the sufficient and necessary conditions on the initial value to the existence of local solution of doubly nonlinear equation. Moreover some results on the global existence and nonexistence of solutions are considered.

Doubly Nonlinear Degenerate Parabolic Equations with a Singular Potential for Greiner Vector Fields

The purpose of this paper is to investigate the nonexistence of positive so lutions of the following doubly nonlinear degenerate parabolic equations:{∂u=▽k·(u^(m-1)|▽ku|^(p-2)▽ku)+(w)u^(m+p-2),u(w,0)=u0(w)≥0,u(w,t)=0,inΩ×(0,T),inΩ,on∂Ω×(0,T),where Q is a Carnot-Carathéodory metric bal in IR^(2n+1)generated by Greiner vector fields,V∈L_(loc)(Ω),k∈N,m∈R,1<p<2n+2k and m+p-2>0.The better lower bound p*for m+p is found and the nonexistence results are proved for p*≤m+p<3.

An Optimal Control Problem of a Coupled Nonlinear Parabolic Population System

An optimal control problem for a coupled nonlinear parabolic population system is considered. The existence and uniqueness of the positive solution for the system is shown by the method of upper and lower solutions. An explicit prior bound of solutions to the system is given by considering an auxiliary coupled linear system. The existence of the optimal control is proved and the characterization of the optimal control is established.

Existence and Uniqueness of Global Solution for Certain Degenerate Reaction-Diffusion Systems

In this paper we discuss the following nonlinear degenerate parabolic systems u_i=△a_i(u_i)+b_i(x,t,u_i)Du_i+f_i(x,t,u)for i = 1,2, …, m and u = (u_1,…, u_m) is a vector function, with Dirichlet boundary condition. Under some structure conditions on a_i,b_i and f_i and initial data u_i^o∈Li(Ω) for some pi>p_i^o = 1,2,…,m, the result on existence and uniquence of global solution is established.