Stanislav Antontsev

Anisotropic parabolic equations with variable nonlinearity

We study the Dirichlet problem for a class of nonlinear parabolic equations with nonstandard anisotropic growth conditions. Equations of this class generalize the evolutional p(x, t)-Laplacian. We prove theorems of existence and uniqueness of weak solutions in suitable Orlicz-Sobolev spaces, derive global and local in time L∞ bounds for the weak solutions.

Chapter 1 Elliptic equations with anisotropic nonlinearity and nonstandard growth conditions

Publisher Summary This chapter discusses the theory of elliptic equations with nonstandard growth conditions and systems of such equations. It discusses the questions of existence, uniqueness, and localization of weak solutions to the formulated problems. The solutions of equations are proved with anisotropic diffusion, which possess a new property of localization caused by strong anisotropy.

Stopping a Viscous Fluid by a Feedback Dissipative Field: Thermal Effects without Phase Changing

We show how the action on two simultaneous effects (a suitable coupling about velocity and temperature and a low range of temperature but upper that the phase changing one) may be responsible of stopping a viscous fluid without any changing phase. Our model involves a system, on an unbounded pipe, given by the planar stationary Navier-Stokes equation perturbed with a sublinear term f(x,θ, u) coupled with a stationary (and possibly nonlinear) advection diffusion equation for the temperature θ.

On the Blow-up of Solutions to Anisotropic Parabolic Equations with Variable Nonlinearity

AbstractThe aim of this paper is to establish sufficient conditions of the finite time blow-up in solutions of the homogeneous Dirichlet problem for the anisotropic parabolic equations with variable nonlinearity

Numerical study of the porous medium equation with absorption, variable exponents of nonlinearity and free boundary

u_t = \sum\nolimits_{i = 1}^n {D_i (a_i (x,t)|D_i u|^{p^i (x) - 2} D_i u) + \sum\nolimits_{i = 1}^K {b_i (x,t)|u|^{\sigma _i (x,t) - 2} u} }

Application of the moving mesh method to the porous medium equation with variable exponent

. Two different cases are studied. In the first case ai ≡ ai(x), pi ≡ 2, σi ≡ σi(x, t), and bi(x, t) ≥ 0. We show that in this case every solution corresponding to a “large” initial function blows up in finite time if there exists at least one j for which min σj(x, t) > 2 and either bj > 0, or bj(x, t) ≥ 0 and Σπbj−ρ(t)(x, t) dx < ∞ with some σ(t) > 0 depending on σj. In the case of the quasilinear equation with the exponents pi and σi depending only on x, we show that the solutions may blow up if min σi ≥ max pi, bi ≥ 0, and there exists at least one j for which min σj > max pj and bj > 0. We extend these results to a semilinear equation with nonlocal forcing terms and quasilinear equations which combine the absorption (bi ≤ 0) and reaction terms.

Elliptic equations with triple variable nonlinearity

In this work, we study the convergence of the finite element method when applied to the following parabolic equation:

Discrete solutions for the porous medium equation with absorption and variable exponents

u_t = div (|u| ^ {\gamma (\mathbf{x})} \nabla u) + f (\mathbf{x}, t)