We establish the existence and multiplicity of solutions for a class of fourth-order superlinear elliptic problems under Navier conditions on the boundary. Here we do not use the Ambrosetti-Rabinowitz condition; instead we assume that the nonlinear term is a nonlinear function which is nonquadratic at infinity.

We prove the existence and uniqueness of global solution for the nonlinear beam equation with initial boundary condition: Q in x f u g t u u M u u ) ( ) ( ) ( ) ( 2 2 2 = ′ + Δ ∇ − Δ + ′ ′ φ α where tt u t x u = ′ ′ ) , ( , t u t x u = ′ ) , ( , 0 > α , φ , , g M is nonlinear functions and Δ is Laplacian in n R .

We study an abstract nonlinear evolution equation governed by time-dependent operator of subdi erential type in a real Hilbert space. In this paper we discuss the convergence of solutions to our evolution equations. Also, we investigate the optimal control problems of nonlinear evolution equations. Moreover, we apply our abstract results to a quasilinear parabolic PDE with a mixed boundary cond...

The asymptotic behavior of global solutions of nonlinear evolution equations, in particular convergence to a certain equilibrium as time tends to infinity has become one of the main concerns in the field of nonlinear evolution equations since 1980s. For one space dimension case, significant progresses have been made (see e.g., [6]). However, the situation in higher space dimension case is much ...

Because of its kinetic nature and computational advantages, the Lattice Boltzmann method (LBM) has been well accepted as a useful tool to simulate micro-scale flows. The slip boundary model plays a crucial role in the accuracy of solutions for micro-channel flow simulations. The most used slip boundary condition is the Maxwell slip model. The results of Maxwell slip model are affected by the ac...

A nonlinear boundary value problem involving the p-biharmonic operator is investigated, where p > 1. It describes various problems in the theory of elasticity, e.g. the shape of an elastic beam where the bending moment depends on the curvature as a power function with exponent p− 1. We prove existence of solutions satisfying a quite general boundary condition that incorporates many particular b...

Abstract. We study a nonlocal diffusion operator in a bounded smooth domain prescribing the flux through the boundary. This problem may be seen as a generalization of the usual Neumann problem for the heat equation. First, we prove existence, uniqueness and a comparison principle. Next, we study the behavior of solutions for some prescribed boundary data including blowing up ones. Finally, we l...

Abstract In this paper, the mixed initial-boundary value problem for inhomogeneous quasilinear strictly hyperbolic systems with nonlinear boundary conditions in the first quadrant {(t, x)| t ≥ 0, x ≥ 0} is investigated. Under the assumption that the right hand side satisfies matching condition and the system is strictly hyperbolic and weakly linearly degenerate, we obtain the global existence a...