On global attractor for m-Laplacian parabolic equation with local and nonlocal nonlinearity

In this paper, we study the long-time behavior of solutions for m-Laplacian parabolic equation in Ω×(0,∞) with the initial data u(x,0)=u₀(x)∈Lq, q?1, and zero boundary condition in ∂Ω. Two cases for a(x)?a₀>0 and a(x)?0 are considered. We obtain the existence and Lp estimate of global attractor A in Lp, for any p?max{1,q}. The attractor A is in fact a bounded set in if a(x)?a₀>0 in Ω, and A is bounded in if a(x)?0 in Ω.     

Global attractor for the m-semiflow generated by a quasilinear degenerate parabolic equation

Using theory of global attractors for multi-valued semiflows, we prove the existence of a global attractor for the m-semiflow generated by a parabolic equation involving the nonlinear degenerate operator in a bounded domain.     

On global behavior of solutions to an inverse problem for nonlinear parabolic equations

We find conditions on data guaranteeing global nonexistence of solutions to an inverse source problem for a class of nonlinear parabolic equations. We also establish a stability result on a bounded domain for a problem with the opposite sign on the power type nonlinearity.     

Growth conditions for oscillation of nonlinear differential equations with p-Laplacian

In this paper, oscillation criteria are established for all solutions of second-order nonlinear differential equations of the form

     

Finite dimensional global attractor for a parametric nonlinear Schrödinger system with a trapping potential

We prove that a parametric nonlinear Schrödinger equation possesses a finite dimensional smooth global attractor in a suitable energy space.     

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.     

A Strong Maximum Principle for parabolic equations with the p-Laplacian

We prove the Strong Maximum Principle (SMP) under suitable assumptions for a class of quasilinear parabolic problems with the p  -Laplacian, p>1$p>1$, on bounded cylindrical domains of R^N+1${\mathbb{R}}^{N+1}$,

_∂tu−_Δpu−λ|u|^p−2u≥0,${\partial }_{t}u-{\mathrm{\Delta }}_{p}u-\lambda {|u|}^{p-2}u\ge 0,$

with nonnegative initial–boundary conditions and λ≤0$\lambda \le 0$, and we give some counterexamples to the SMP if some of our assumptions are violated. We show that the Hopf Maximum Principle holds for 1<p<2$1<p<2$, and give a counterexample to it for p>2$p>2$. Also the Weak Maximum Principle for λ≤_λ1$\lambda \le {\lambda }_1$ is established.     

Periodic solutions of the evolutionary p-Laplacian with nonlinear sources

This paper is concerned with the evolutionary p-Laplacian with nonlinear and periodic sources. We will give a rather complete characterization, in terms of the parameter p and the exponent q of the source, of whether or not the positive periodic solutions exist.     

Finite dimensional global attractor for a class of doubly nonlinear parabolic equations

Our aim in this paper is to study the long time behavior of a class of doubly nonlinear parabolic equations. In particular, we prove the existence of the global attractor which has, in one and two space dimensions, finite fractal dimension.     

The existence of multiple equilibrium points in a global attractor for some p-Laplacian equation

In this paper, we are mainly concerned with some properties of the global attractor for some p-Laplacian equation with a Lyapunov function in a Banach space. Under some suitable assumptions, we prove the existence of multiple equilibrium points in a global attractor for some p-Laplacian equation.     

Blow-up and global solutions for a class of nonlinear parabolic equations with different kinds of boundary conditions

By constructing different auxiliary functions and using Hopf’s maximum principle, the sufficient conditions for the blow-up and global solutions are presented for nonlinear parabolic equation u_t = ∇(a(u)b(x)c(t)∇u) + f(x, u, q, t) with different kinds of boundary conditions. The upper bounds of the “blow-up time” and the “upper estimates” of global solutions are provided. Finally, some examples are presented as the application of the obtained results.     

Oscillation criteria for second-order damped nonlinear differential equations with p-Laplacian

This paper is concerned with the oscillation problem for the nonlinear differential equation with a damping term,

     

Global attractor for a class of nonlinear lattices

We consider a class of nonlinear lattices with nonlinear damping

(0.1)     

Regularity in Sobolev spaces for doubly nonlinear parabolic equations

The doubly nonlinear parabolic equation

     

Localization for the evolution p-Laplacian equation with strongly nonlinear source term

In this paper we study the strict localization for the p-Laplacian equation with strongly nonlinear source term. Let u:=u(x,t) be a solution of the Cauchy problem

     

Multiple solutions for coercive p-Laplacian equations

We obtain multiple nonzero solutions for coercive p-Laplacian equations. In order to obtain the third nonzero solution, we use the second deformation lemma to construct the desired mountain pass path.     

Landesman-Lazer type conditions for a system of p-Laplacian like operators

We study the existence of periodic solutions for a nonlinear second order system of ordinary differential equations of p-Laplacian type. Assuming suitable Nagumo and Landesman-Lazer type conditions we prove the existence of at least one solution applying topological degree methods. We extend a celebrated result by Nirenberg for resonant systems.     

Renormalized solutions of nonlinear parabolic equations with general measure data

Let a bounded open set, N ≥  2, and let p > 1; we prove existence of a renormalized solution for parabolic problems whose model is