Sharp criteria of global existence and blow-up for a type of nonlinear parabolic equations

We study a type of nonlinear parabolic equations. In terms of the variational characterization of the corresponding nonlinear elliptic equations and the invariant flow arguments, we establish the sharp criteria for global existence and blow-up. Furthermore, we also get the instability of the steady states and the global existence with small initial data.     

Fujita type exponent for fully nonlinear parabolic equations and existence results

In this paper, we prove that a class of parabolic equations involving a second order fully nonlinear uniformly elliptic operator has a Fujita type exponent. These exponents are related with an eigenvalue problem in all RN and play the same role in blow-up theorems as the classical p^?=1+2/N introduced by Fujita for the Laplacian. We also obtain some associated existence results.     

Blow-up and global existence to a degenerate reaction-diffusion equation with nonlinear memory

In this paper, we consider a degenerate reaction-diffusion equation

     

Oscillation of fourth order nonlinear neutral difference equations I

Oscillatory and asymptotic behaviour of solutions of a class of nonlinear fourth order neutral difference equations of the form

Heisenberg群上四阶非线性次椭圆方程的可解性

§ 1　ＩｎｔｒｏｄｕｃｔｉｏｎＷｅｃｏｎｓｉｄｅｒｔｈｅｆｏｕｒｔｈｏｒｄｅｒｓｅｍｉｌｉｎｅａｒｓｕｂｅｌｌｉｐｔｉｃｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌｅｍΔ2 Ｈｕ +ｃΔＨｕ =ｆ( (ｚ ,ｔ) ,ｕ)　ｉｎＤ ,ｕ｜Ｄ =ΔＨｕ｜Ｄ =0 ,( 1 .1 )ｗｈｅｒｅＤｉｓａｂｏｕｎｄｅｄｏｐｅｎｓｕｂｓｅｔｏｆｔｈｅＨｅｉｓｅｎｂｅｒｇｇｒｏｕｐＨｎａｎｄΔＨｉｓｔｈｅｓｕｂｅｌｌｉｐｔｉｃＬａｐｌａ ｃｉａｎｏｎＨｎ.ＷｅｒｅｃａｌｌｔｈａｔＨｎｉｓｔｈｅＬｉｅｇｒｏｕｐｗｈｏｓｅｕｎｄｅｒｌｙｉｎｇｍａｎｉ…     

A class of fourth order wave equations with dissipative and nonlinear strain terms

We study the initial boundary value problem for a class of fourth order wave equations with dissipative and nonlinear strain terms. By introducing a family of potential wells we not only obtain the invariant sets and vacuum isolating of solutions, but also give some threshold results of global existence and nonexistence of solutions.     

Higher order equations related to thin films: Blow-up and global existence, the influence of the initial data

We are interested in the following class of equations:

     

Global existence and boundedness for a class of second-order nonlinear differential equations

In this paper we obtain new conditions for the global existence and boundedness of solutions for nonlinear second-order equations of the form ${\left(r\left(t\right){\left|u^{\prime }\right|}^{p?2}{u}^{\prime }\right)}^{\prime }+g(t,u,u^{\prime })u^{\prime }+a\left(t\right)f\left(u\right)=e\left(t\right)\text{,}$ where $p>1$ is a real constant. The results are applicable to well-known Emden–Fowler and Lienard type equations. An illustrative example is also provided.     

Low regularity global solutions for nonlinear evolution equations of Kirchhoff type

We study the global solvability of the Cauchy-Dirichlet problem for two second order in time nonlinear integro-differential equations:

1)

the extensible beam/plate equation

     

Finite time blow-up and global existence of weak solutions for pseudo-parabolic equation with exponential nonlinearity

This paper is concerned with the initial boundary value problem of a class of pseudo-parabolic equation $u_t - \triangle u - \triangle u_t + u = f(u)$ with an exponential nonlinearity. The eigenfunction method and the Galerkin method are used to prove the blow-up, the local existence and the global existence of weak solutions. Moreover, we also obtain other properties of weak solutions by the eigenfunction method.     

A fourth order iterative method for solving nonlinear equations

The present paper illustrates an iterative numerical method to solve nonlinear equations of the form f(x) = 0, especially those containing the partial and non partial involvement of transcendental terms. Comparative analysis shows that the present method is faster than Newton-Raphson method, hybrid iteration method, new hybrid iteration method and others. Cost is also found to be minimum than these methods. The beauty in our method can be seen because of the optimization in important effecting factors, i.e. lesser number of iteration steps, lesser number of functional evaluations and lesser value of absolute error in final as well as in individual step as compared to the other methods. This work also demonstrates the higher order convergence of the present method as compared to others without going to the computation of second derivative.     

Existence and nonexistence of a global solution for coupled nonlinear wave equations with damping and source

We consider the system of nonlinear wave equations

     

Estimates of low Sobolev norms of solutions for some nonlinear evolution equations

In this work we obtain results on the estimates of low Sobolev norms for solutions of some nonlinear evolution equations, in particular we apply our method for the complex modified Korteweg-de Vries type equation and Benjamin-Ono equation.     

Global existence of solution of Cauchy problem for nonlinear pseudo-parabolic equation

In this paper, we prove that the Cauchy problem for the nonlinear pseudo-parabolic equation

vt−αvxxt−βvxx+γvx+fx(v)=φx(vx)+g(v)−αg(v)_xx     

Global existence and nonexistence of solution for Cauchy problem of two-dimensional generalized Boussinesq equations

In this paper we consider the Cauchy problem of two-dimensional generalized Boussinesq-type equation _utt−Δu−Δ_utt+Δ²u+Δf(u)=0$u_{tt}-\mathrm{\Delta }u-\mathrm{\Delta }{u}_{tt}+{\mathrm{\Delta }}^{2}u+\mathrm{\Delta }f(u)=0$. Under the assumption that f(u)$f(u)$ is a function with exponential growth at infinity and under some assumptions on the initial data, we prove the existence and nonexistence of global weak solution. There are very few works on Boussinesq equation with nonlinear exponential growth term by potential well theory.     

On a solution to fractional order integrodifferential equations with analytic semigroups

In this paper we study a class of fractional order integrodifferential equations considered in an arbitrary Banach space. Using the theory of analytic semigroups we establish the existence, uniqueness and regularity of a mild solution to these fractional order integrodifferential equations.     

On the existence of positive solutions of nonlinear second order differential equations

Under suitable conditions on , the boundary value problem

has at least one positive solution. Moreover, we also apply this main result to establish several existence theorems of multiple positive solutions for some nonlinear (elliptic) differential equations.