Sergey Shmarev

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.

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

Elliptic equations with triple variable nonlinearity

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} }

The Function Spaces

. 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.

On the Boundary Layer for Dilatant Fluids

with continuous compactly supported initial data u(x, 0) = u0(x) 0 in the critical case m + p = 2 of the range of parameters m > 1, p 1) and strong absorption ( p < 1).

On the localization of solutions of doubly nonlinear parabolic equations with nonstandard growth in filtration theory

We study the homogeneous Dirichlet problem for the elliptic equation with triple variable nonlinearity where the exponents α(x) and p(x) are given functions, Ω ⊂ ℝ n is a domain with Lipschitz-continuous boundary. We show that under suitable conditions on the exponents α(x) and p(x), and on the growth of the term f(x, u), the problem has a bounded weak solution in the Orlicz–Sobolev space. The uniqueness is proved under the assumption that the mapping s ↦ f(x, s) is nondecreasing and a(x, u) ≡ a(x). We give sufficient conditions for the dead core formation in the special case f(x, u) = c(x, u)|u|σ(x)−2 u + Φ(x) with c(x, u) ≥ c − > 0 and σ(x) > 1 in Ω.

On a class of fully nonlinear parabolic equations

ABSTRACT We study the global regularity of solutions of the homogeneous Dirichlet problem for the parabolic equation with variable nonlinearity where p(x, t), are given functions of their arguments, and . Conditions on the data are found that guarantee the existence of a unique strong solution such that and . It is shown that if with , p and are Hölder-continuous in , and , then for every strong solution with any .