Existence of Weak Solutions for Nonlinear Time-Fractionalp-Laplace Problems

The existence of weak solution forp-Laplace problem is studied in the paper. By exploiting the relationship between the Nehari manifold and fibering maps and combining the compact imbedding theorem and the behavior of Palais-Smale sequences in the Nehari manifold, the existence of weak solutions is established. By means of the Arzela-Ascoli fixed point theorem, some existence results of the corresponding time-fractional equations of thep-Laplace problem are obtained.

Existence of Minimizer to the Ginzburg-Landau Free Energy of Ferromagnetic System

In this paper, we obtain the existence of minimizer to Ginzburg-Landau free energy of ferromagnetic system by coercivity and weakly lower semi-continuity of the free energy, where the weakly lower semi-continuity is derived from monotone operator condition and the Sobolev space compact imbedding theorem.

Uniform Attractor and Approximate Inertial Manifolds for Nonautonomous Long-Short Wave Equations

Nonautonomous long-short wave equations with quasiperiodic forces are studied. We prove the existence of the uniform attractor for the system by means of energy method, which is widely used to deal with problems who have no continuity (with respect to the initial data) property, as well as to those which Sobolev compact imbedding cannot be applied. Afterwards, we construct an approximate inertial manifold by means of extending phase space method and we estimated the size of the corresponding attracting neighborhood for this manifold.

On a Nonlinear Wave Equation Associated with Dirichlet Conditions: Solvability and Asymptotic Expansion of Solutions in Many Small Parameters

A Dirichlet problem for a nonlinear wave equation is investigated. Under suitable assumptions, we prove the solvability and the uniqueness of a weak solution of the above problem. On the other hand, a high-order asymptotic expansion of a weak solution in many small parameters is studied. Our approach is based on the Faedo-Galerkin method, the compact imbedding theorems, and the Taylor expansion of a function.