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This article presents suffcient conditions for the existence of solutions of the anisotropic quasilinear elliptic equation with variable exponent and nonlinear Robin boundary conditions, -ΣNi=1φ/φxi (|φu/φxi)NΣi=1|u|pi(x)-2u + λ|u|m(x)-2u = γg(x,u) in Ω NΣi=1|φu/φxi|pi(x)-2 φu/φxi vi = μ|u|q(x-2)u on ∂Ω Under appropriate assumptions on the data, we prove some existence and multiplicity results. The methods are based on Mountain Pass and Fountain theorems. [ABSTRACT FROM AUTHOR]

We study the doubly nonlinear parabolic equation ∂β(u)/∂t - Δpu + f(x, t, u) = 0 in Ω x R+, with Dirichlet boundary conditions and initial data. We investigate a time-discretization of the continuous problem by the Euler forward scheme. In addition to proving existence, uniqueness and stability questions, we study the long time behavior of the solution to the discrete problem. We prove the existence of a global attractor, and obtain its regularity under additional conditions. [ABSTRACT FROM AUTHOR]