Evolution PDEs with Nonstandard Growth Conditions: by Stanislav Antontsev, Sergey Shmarev

By Stanislav Antontsev, Sergey Shmarev

This monograph bargains the reader a remedy of the idea of evolution PDEs with nonstandard progress stipulations. This type comprises parabolic and hyperbolic equations with variable or anisotropic nonlinear constitution. We enhance equipment for the research of such equations and current a close account of contemporary effects. an outline of alternative ways to the examine of PDEs of this sort is equipped. The presentation is targeted at the problems with life and specialty of strategies in applicable functionality areas and at the examine of the explicit qualitative homes of strategies, similar to localization in house and time, extinction in a finite time and blow-up, or nonexistence of worldwide in time recommendations. specified consciousness is paid to the examine of the homes intrinsic to options of equations with nonstandard growth.

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Additional resources for Evolution PDEs with Nonstandard Growth Conditions: Existence, Uniqueness, Localization, Blow-up

Example text

If p(x), q(x) ∈ C(Ω) and q(x) < p ∗ (x) in Ω, then for every u ∈ W0 (Ω) u q(·),Ω ≤ C ∇u p(·),Ω with a constant C depending on p ± , n, the properties of ∂Ω and the modulus of con1, p(·) tinuity of p(x). The embedding W0 (Ω) ⊂ L q(·) (Ω) is continuous and compact. Proof Let us consider first the case p + < n. For an arbitrary fixed x ∈ Ω there is a neighborhood Ux such that min p(x) max q(x) < Ux Ux n − min p(x) Ux Let {Ux }x∈Ω be an open covering of the compact Ω. Choose a finite subcover {Ui : i = 1, 2, .

16) by u − , integrating over Ω and taking into account the equalities u − (x, 0) = 0, u − Γ = 0 we find that T 1 d u − (·, t) 2 dt 2,Ω + Ω a|∇u − |2 d x ≤ 0. 50 2 A Porous Medium Equation with Variable Nonlinearity It follows that for every t > 0 0 ≤ u − (·, t) ≤ u − (x, 0) 2,Ω = 0, 2,Ω whence the assertion. 20) with an independent of ε constant C. 18) with k = 1 that for every t ∈ (0, T ) and τ ∈ [0, 1] 1 u(·, t) 2 2 2,Ω + a |∇u|2 d xdt ≤ Qt τ u0 2 T 1 u 0 22,Ω + K (T ) 2 0 ≤ (|Ω| + K (T ))K (T ).

21) where ω is a continuous function such that lim ω(τ ) ln τ →0+ 1 = C < +∞, C = const. τ Let ρ(x) be the Friedrichs mollifying kernel ρ(x) = κ exp −1 1−|x|2 if |x| < 1, κ = const such that if |x| > 1, 0 Denote ρε (x) = ε−n ρ x , ε Rn ρ(x) d x = 1. ε > 0. 1, p(·) Given f ∈ W0 (Ω), we continue it by zero to the whole of Rn and use the same notation for the continued function. Let us consider the sequence of functions f ε (x) = f ∗ ρε ≡ Rn f (y) ρε (y − x) dy, ε > 0. 22) It is clear that since Ω is bounded, there a ball B R (0) = {x ∈ Rn : |x| < R} such that Ω ⊂ B R (0), and that supp f ε ⊂ B R+1 (0) ≡ B for all ε ∈ (0, 1).

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