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Stability of solutions to abstract evolution equations with delay
 J. Math. Anal. Appl
"... Abstract An equationu = A(t)u+B(t)F (t, u(t−τ )), u(t) = v(t), −τ ≤ t ≤ 0 is considered, A(t) and B(t) are linear operators in a Hilbert space H, u = du dt , F : H → H is a nonlinear operator, τ > 0 is a constant. Under some assumption on A(t), B(t) and F (t, u) sufficient condittions are given ..."
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Abstract An equationu = A(t)u+B(t)F (t, u(t−τ )), u(t) = v(t), −τ ≤ t ≤ 0 is considered, A(t) and B(t) are linear operators in a Hilbert space H, u = du dt , F : H → H is a nonlinear operator, τ > 0 is a constant. Under some assumption on A(t), B(t) and F (t, u) sufficient condittions are given for the solution u(t) to exist globally, i.e, for all t ≥ 0, to be globally bounded, and to tend to zero as t → ∞. MSC: 34G20, 34K20, 37L05, 47J35
Convergence of TimeDependent Turing Structures to a Stationary Solution
"... Abstract. Stability of stationary solutions of parabolic equations is conventionally studied by linear stability analysis, Lyapunov functions or lower and upper functions. We discuss here another approach based on differential inequalities written for the L 2 norm of the solution. This method is app ..."
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Abstract. Stability of stationary solutions of parabolic equations is conventionally studied by linear stability analysis, Lyapunov functions or lower and upper functions. We discuss here another approach based on differential inequalities written for the L 2 norm of the solution. This method is appropriate for the equations with time dependent coefficients. It yields new results and is applicable when the usual linearization method is not applicable. Key words: parabolic systems, stationary solutions, stability, differential inequalities
Stability of Solutions to Evolution Problems
, 2013
"... Large time behavior of solutions to abstract differential equations is studied. The results give sufficient condition for the global existence of a solution to an abstract dynamical system (evolution problem), for this solution to be bounded, and for this solution to have a finite limit as t → ∞, ..."
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Large time behavior of solutions to abstract differential equations is studied. The results give sufficient condition for the global existence of a solution to an abstract dynamical system (evolution problem), for this solution to be bounded, and for this solution to have a finite limit as t → ∞, in particular, sufficient conditions for this limit to be zero. The evolution problem is: ˙u = A(t)u + F (t, u) + b(t), t ≥ 0; u(0) = u0. (∗) Here ˙u: = du, u = u(t) ∈ H, H is a Hilbert space, t ∈ R+:= dt [0, ∞), A(t) is a linear dissipative operator: Re(A(t)u, u) ≤ −γ(t)(u, u), where F (t, u) is a nonlinear operator, ‖F (t, u) ‖ ≤ c0‖u‖p, p> 1, c0 and p are positive constants, ‖b(t) ‖ ≤ β(t), and β(t) ≥ 0 is a continuous function. The basic technical tool in this work are nonlinear differential inequalities. The nonclassical case γ(t) ≤ 0 is also treated.