S. Antontsev Universidad da Beira Interior, Covilhã, Portugal

and

S. Shmarev Universidad de Oviedo, Spain

Parabolic equations with anisotropic non-uniform degeneracy

Let Ω ⊂ ℝ^{n} be a bounded domain with Lipschitz-continuous boundary Γ and Q = Ω × (0, T]. We study the parabolic equations with anisotropic non-uniform degeneracy

u_{t}−∑_{i=1}^{n} D_{i} (a_{i}(x, t, u)|D_{i}u|^{p_{i}(x,t)−2}D_{i}u)+c(x, t, u)|u|^{σ(x,t)−2}u = f(x, t) in Q,

(1)

u_{t}−∑_{i=1}^{n} D_{i} (a_{i}(x, t, u)|u|^{γ_{i}(x,t)}D_{i} u + b_{i}(x, t, u)|u|^{δ_{i}(x,t)}u)+c(x, t, u)|u|^{σ(x,t)}u = f(x, t)

(2)

under the boundary and initial conditions

u = 0 on Γ,  u(x, 0) = u_{0}(x) in Ω.  (3)

The coefficients γ_{i}, δ_{i}, a_{i}, b_{i}, c, p_{i} and σ are given functions of their arguments. Such equations emerge from the mathematical modelling of electrorheological fluids, fluids with temperature-dependent viscosity, the processes of filtration in inhomogenenous anisotropic media. It is assumed that the coefficients a_{i}, c are bounded, a_{i} > 0, c ≥ 0, and the exponents p_{i}(x, t) > 1, σ(x, t) > 1 are continuous in Q with a logarithmic module of continuity.

We prove the existence and uniqueness of weak solutions of problems (??), (??) and (??), (??) and study the localization (vanishing) properties of weak solutions and the effect of finite time stabilization of solutions to a stationary profile. The study of the localization properties is performed with the method of local energy estimates [?]. The detailed proofs can be found in [?, ?, ?, ?]

References

[1] S. Antontsev and V. Zhikov, Higher integrability for parabolic equations of p(x, t)− Laplacian type, Preprint 5, Universidade da Beira Interior, Portugal, www.ubi.pt/externos/noe.html, (2004), pp. 0–30.

[2] S. N. Antontsev, J. I. Díaz, and S. Shmarev, Energy Methods for Free Boundary Problems:Applications to Non-linear PDEs and Fluid Mechanics, Bikhäuser, Boston, 2002. Progress in Nonlinear Differential Equations and Their Applications, Vol. 48.

[3] S. N. Antontsev and F. Rodrigues, On stationary thermorheological viscous flows, Ann. Univ. Ferrara, Sez. , Sc. Mat. To appear.

[4] S. N. Antontsev and S. I. Shmarev, Existence and uniqueness of solutions of degenerate parabolic equations with variable exponents of nonlinearity, Preprint 3, Universidade da Beira Interior, Portugal, www.ubi.pt/externos/noe.html, (2004), pp. 1–16.

[5] , A model porous medium equation with variable exponent of nonlinearity: existence, uniqueness and localization properties of solutions, Journal of Nonlinear Analysis, 60 (2005), pp. 515–545.