Regularity of Weak Solutions and Their Attractors for a Parabolic Feedback Control Problem

Lyapunov Functions for Weak Solutions of Reaction-Diffusion Equations with Discontinuous Interaction Functions and its Applications

“…Then, there exists C > 0 such that for any τ < T and for each weak solution u(·) of problem (1.1)-(1.2) on [τ, T] the inequality holds(t − τ)‖u(t)‖ 2 V + t ∫︁ τ (s − τ)‖u(s)‖ 2 H 2 (Ω)∩V ds ≤ C(1 + ‖u(τ)‖ 2 H + (t − τ) 2 ) ∀t ∈ (τ, T].Remark 2.4. The proof of Theorem 2.3 is similar to the proof of Theorem 2 from Kasyanov et al[28], however it was proved under another assumptions on the interaction function.Brought to you by | University of New Orleans Authenticated Download Date | 6/1/15 1:37 PMProof. Let τ < T and u(·) be an arbitrary weak solution of problem (1.1)-(1.2) on [τ, T].…”

Uniform Trajectory Attractors for Nonautonomous Dissipative Dynamical Systems

“…For applications, one can consider new classes of problems with degeneracy, feedback control problems, problems on manifolds, problems with delay, stochastic partial differential equations, etc. (see Balibrea et al [2]; Hu and Papageorgiou [3]; Gasinski and Papageorgiou [5]; Kasyanov [15]; Kasyanov, Toscano, and Zadoianchuk [17]; Mel'nik and Valero [21]; Denkowski, Migórski, and Papageorgiou [13]; Gasinski and Papageorgiou [5]; Zgurovsky et al [29,30]; etc.) involving differential operators of pseudomonotone type and the corresponding choice of the phase spaces.…”

Lyapunov Functions for Differential Inclusions and Applications in Physics, Biology, and Climatology

“…Topological properties of strong and weak solutions were considered in [14,[31][32][33][34]. Regularity properties of global and trajectory attractors were provided in [18,[24][25][26] (14.5), is a Lyapunov type function for K + . Moreover, for each u ∈ K + and all τ and T , 0 < τ < T < ∞, the energy equality holds…”

“…We recall that the multivalued map G : R + ×H → P(H ) is said to be a strict multivalued semiflow (strict m-semiflow) if: G(s, x)) ∀x ∈ H, t, s ∈ R + . We recall that the set A ⊆ H is said to be an invariant global attractor of G if: Let {T (h)} h≥0 be the translation semigroup acting on A set U ⊂ K + is said to be trajectory attractor in the trajectory space K + with respect to the topology of C loc (R + ; H ), if U ⊂ K + is a global attractor for the translation semigroup {T (h)} h≥0 acting on K + ; Kasyanov et al [26,Sect. 3].…”