EXISTENCE AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR NONLINEAR PARABOLIC EQUATIONS WITH VARIABLE EXPONENT OF NONLINEARITY

The authors of this article study the existence and uniqueness of weak so- lutions of the initial-boundary value problem for ut = div（（｜u｜^δ ＋ d0）｜↓△｜^p（x,t）-2↓△u） ＋ f（x, t） （0 〈 δ 〈 2）. They apply the method of parabolic regularization and Galerkin＇s method to prove the existence of solutions to the mentioned problem and then prove the uniqueness of the weak solution by arguing by contradiction. The authors prove that the solution approaches 0 in L^2 （Ω） norm as t →∞.

OPTIMAL CONDITIONS OF GLOBAL EXISTENCE AND BLOW-UP FOR A NONLINEAR PARABOLIC EQUATION

According to the variational analysis and the potential well argument, we get the optimal conditions of global existence and blow-up for a type of nonlinear parabolic equations. Furthermore, we give its application in the instability of the steady states.

GRADIENT ESTIMATES AND LIOUVILLE THEOREMS FOR LINEAR AND NONLINEAR PARABOLIC EQUATIONS ON RIEMANNIAN MANIFOLDS

In this article, we will derive local elliptic type gradient estimates for positive solutions of linear parabolic equations （△-e/et）u（x,t）＋q（x,t）u^p（x,t）=0 and nonlinear parabolic equations （△-e/et）u（x,t）＋h（x,t）u^p（x,t）=0（p 〉 1） on Riemannian manifolds.As applications, we obtain some theorems of Liouville type for positive ancient solutions of such equations. Our results generalize that of Souplet-Zhang （[1], Bull. London Math. Soc. 38（2006）, 1045-1053） and the author （[2], Nonlinear Anal. 74 （2011）, 5141-5146）.

Hamilton-Souplet-Zhang's gradient estimates for two weighted nonlinear parabolic equations

In this paper, we consider gradient estimates for positive solutions to the following weighted nonlinear parabolic equations on a complete smooth metric measure space with only Bakry-émery Ricci tensor bounded below: One is $${u_t} = {\Delta _f}u + au\log u + bu$$ with a, b two real constants, and another is $${u_t} = {\Delta _f}u + \lambda {u^\alpha }$$ with λ, α two real constants. We obtain local Hamilton-Souplet-Zhang type gradient estimates for the above two nonlinear parabolic equations. In particular, our estimates do not depend on any assumption on f.

Extinction of Weak Solutions for Nonlinear Parabolic Equations with Nonstandard Growth Conditions

This paper deals with the extinction of weak solutions of the initial and boundary value problem for ut = div（（｜u｜σ ＋ d0）｜ u｜^p（x）-2 u）. When the exponent belongs to different intervals, the solution has different singularity （vanishing in finite time）.

ON GLOBAL EXISTENCE AND NONEXISTENCE FOR SOME NONLINEAR PARABOLIC EQUATIONS

This paper gives the sufficient and necessary conditions of existence of global solutions and decay estimates of the solutions for the initial boundary value problem of some nonlinear parabolic equations with small initial energy and the nonlinear power less than Sobolev critical value. The existence, nonexistence and the decay estimates of global solutions are considered. The conditions that initial energy is small and nonlinear power is less than Sobolev critical value is imposed.

Local Hamilton type Gradient Estimates and Harnack Inequalities for Nonlinear Parabolic Equations on Riemannian Manifolds

In this paper,we prove a local Hamilton type gradient estimate for positive solution of the nonlinear parabolic equation ut(x,t)=Δu(x,t)+au(x,t) ln u(x,t)+bu^(α)(x,t),on M×(-∞,∞) with α∈R,where a and b are constants.As application,the Harnack inequalities are derived.

Global Existence and Extinction Behaviour for a Doubly Nonlinear Parabolic Equation with Logarithmic Nonlinearity

This paper is mainly focused on the global existence and extinction behaviour of the solutions to a doubly nonlinear parabolic equation with logarithmic nonlinearity. By making use of energy estimates method and a series of ordinary differential inequalities, the global existence of the solution is obtained. Moreover, we give the sufficient conditions on the occurrence(or absence)of the extinction behaviour.

ACCELERATION METHODS OF NONLINEAR ITERATION FOR NONLINEAR PARABOLIC EQUATIONS

This paper discusses the accelerating of nonlinear parabolic equations. Two iterative methods for solving the implicit scheme new nonlinear iterative methods named by the implicit-explicit quasi-Newton （IEQN） method and the derivative free implicit-explicit quasi-Newton （DFIEQN） method are introduced, in which the resulting linear equations from the linearization can preserve the parabolic characteristics of the original partial differential equations. It is proved that the iterative sequence of the iteration method can converge to the solution of the implicit scheme quadratically. Moreover, compared with the Jacobian Free Newton-Krylov （JFNK） method, the DFIEQN method has some advantages, e.g., its implementation is easy, and it gives a linear algebraic system with an explicit coefficient matrix, so that the linear （inner） iteration is not restricted to the Krylov method. Computational results by the IEQN, DFIEQN, JFNK and Picard iteration methods are presented in confirmation of the theory and comparison of the performance of these methods.

The Harnack Estimate for a Nonlinear Parabolic Equation under the Ricci Flow

Let （M,g（t））, 0 ≤ t ≤ T, be an n-dimensional closed manifold with nonnegative Ricci c for some constant C 〉 0 and g（t） evolving by the Ricci flow curvature, │Rc│ ≤C/t for some constant C 〉 0 and g（t） evolving by the Ricci flow gij/ t=-2Rij.In this paper, we derive a differential Harnack estimate for positive solutions to parabolic equations of the type u~ = /△u - aulogu - bu on M x （0,T], where a 〉 0 and b ∈ R.

Gradient Estimates and Harnack Inequality for a Nonlinear Parabolic Equation on Complete Manifolds

Let M be a noncompact complete Riemannian manifold.In this paper,we consider the following nonlinear parabolic equation on M ut(x,t)=△u(x,t)+au(x,t)ln u(x,t)+bu^α(x,t).We prove a Li–Yau type gradient estimate for positive solutions to the above equation;as an application,we also derive the corresponding Harnack inequality.These results generalize the corresponding ones proved by Li(J Funct Anal 100:233–256,1991).

Analysis of Two-Grid Methods for Nonlinear Parabolic Equations by Expanded Mixed Finite Element Methods

In this paper,we present an efficient method of two-grid scheme for the approximation of two-dimensional nonlinear parabolic equations using an expanded mixed finite element method.We use two Newton iterations on the fine grid in our methods.Firstly,we solve an original nonlinear problem on the coarse nonlinear grid,then we use Newton iterations on the fine grid twice.The two-grid idea is from Xu's work[SIAM J.Numer.Anal.,33(1996),pp.1759–1777]on standard finite method.We also obtain the error estimates for the algorithms of the two-grid method.It is shown that the algorithm achieve asymptotically optimal approximation rate with the two-grid methods as long as the mesh sizes satisfy h=O(H^((4k+1)/(k+1))).

Nonlinear Parabolic Equations with Singularities in Colombeau Vector Spaces

We consider nonlinear parabolic equations with nonlinear non-Lipschitz＇s term and singular initial data like Dirac measure, its derivatives and powers. We prove existence-uniqueness theorems in Colombeau vector space yC^1,W^2,2（[0,T）,R^n）,n ≤ 3. Due to high singularity in a case of parabolic equation with nonlinear conservative term we employ the regularized derivative for the conservative term, in order to obtain the global existence-uniqueness result in Colombeau vector space yC^1,L^2（[0,T）,R^n）,n≤ 3.

STRONGLY NONLINEAR VARIATIONAL PARABOLIC EQUATIONS WITH p(x)-GROWTH

We study the Dirichlet problem associated to strongly nonlinear parabolic equations involving p（x） structure in W;L;（Q）. We prove the existence of weak solutions by applying Galerkin’s approximation method.

Nonlinear Parabolic Equations with Singular Coefficient with Respect to the Unknown and with Diffuse Measure Data

An existence and uniqueness result of a renormalized solution for a class of doubly nonlinear parabolic equations with singular coefficient with respect to the unknown and with diffuse measure data is established.A comparison result is also proved for such solutions.

Global Solution for Fully Nonlinear Parabolic Equations in One-Dimensional Space

We discuss the existence of global classical solution for the uniformly parabolic equation ■ut=a(x,t,u,ux,uxx)+b（x,t,u,ux）,（x,t）∈（-1,1）×(0,T], u（±1,t）=0,u（x,0）=■（x）, where a is strongly nonlinear with respect to uxxand ■ is not necessarily small.We also deal with nonuniform case.

INITIAL VALUE PROBLEM FOR A NONLINEAR PARABOLIC EQUATION WITH SINGULAR INTEGRAL-DIFFERENTIAL TERM

We study the initial value problem for a nonlinear parabolic equation with singular integral-differential term. By means of a series of a priori estimations of the solutions to the problem andLeray-Schauder fixed point principle, we demonstrate the existence and uniqueness theorems ofthe generalized and classical global solutions. Lastly, we discuss the asymptotic properties of thesolution as t tends to infinity.

A lumped mass nonconforming finite element method for nonlinear parabolic integro-differential equations on anisotropic meshes

A lumped mass approximation scheme of a low order Crouzeix-Raviart type noncon- forming triangular finite element is proposed to a kind of nonlinear parabolic integro-differential equations. The L2 error estimate is derived on anisotropic meshes without referring to the traditional nonclassical elliptic projection.

Fully Nonlinear Parabolic Equations and the Dini Condition

Interior regularity results for viscosity solutions of fully nonlinear uniformly parabolic equations under the Dini condition, which improve and generalize a result due to Kovats, are obtained by the use of the approximation lemma.

ON THE CAUCHY PROBLEM OF NONLINEAR DEGENERATE PARABOLIC EQUATION

In this paper, we prove the existence of solution of the Cauchy problem of a nonlinear degenerate parabolic equation. Moreover some regularizing effects are exhibited.