## Parabolic Evolution Equations With Asymptotically Autonomous Delay (2001)

Venue: | Report No.2, Fachbereich Mathematik und Informatik, Universitat |

Citations: | 3 - 3 self |

### BibTeX

@TECHREPORT{Schnaubelt01parabolicevolution,

author = {Roland Schnaubelt},

title = {Parabolic Evolution Equations With Asymptotically Autonomous Delay},

institution = {Report No.2, Fachbereich Mathematik und Informatik, Universitat},

year = {2001}

}

### OpenURL

### Abstract

. We study retarded parabolic non--autonomous evolution equations whose coefficients converge as t ! 1 such that the autonomous problem in the limit has an exponential dichotomy. Then the non--autonomous problem inherits the exponential dichotomy and the solution of the inhomogeneous equation tends to the stationary solution at infinity. We use a generalized characteristic equation to deduce the exponential dichotomy and new representation formulas for the solution of the inhomogeneous equation. 1. Introduction In the present paper we continue the investigation of the long--term behaviour of asymptotically autonomous evolution equations begun in [30]. There we studied the Cauchy problem u(t) = A(t)u(t) + f(t); t ? s 0; u(s) = x; (1.1) on a Banach space X assuming that the linear operators A(t), t 0, are sectorial of the same type and satisfy the `Acquistapace--Terreni' condition (see (P) below) and that there exists another sectorial operator A such that lim t!1 R(w; A(t...

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Citation Context ...ur methods are inspired by the evolution semigroup approach and by the papers [12], [13], [14], [24]. Several auxiliary results are presented in the next section. Unexplained notation can be found in =-=[10]-=-. By c we denote a generic constant. 2. Prerequisites Parabolic evolution equations. Let X be a Banach space and J be either R or [a; 1). A collection fU(t; s) : tss; t; s 2 Jg ` L(X) is said to be an... |

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Citation Context ...ch that U(\Delta; s) 2 C 1 ((s; 1);L(X)) and @ t U(t; s) = A(t)U(t; s) for t ? s. Moreover, u = U(\Delta; s)x 2 C 1 ([s; 1);X) is the unique solution of (1.1) for x 2 D(A(s)) and f = 0, see also [2], =-=[3]-=-, [34], [35]. We say that A(\Delta) generates U(\Delta; \Delta). This evolution family also possesses parabolic 3 regularity in the sense that kU(t; s)k L(X s ` ;X t ` ) + (t \Gamma s) ` kU(t; s)k L(X... |

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Citation Context ...d in [10, xVI.6], see also [33, x3.1, 3.2] and the references therein. In this paper we extend our previous results from the undelayed case to the setting introduced above, see [7], [17, x9.5], [20], =-=[22]-=- for related investigations if X = C n . In Theorem 3.6 we establish that the homogeneous problem (1.5) with f = 0 has an exponential dichotomy if condition (1.9) holds. This theorem relies on Proposi... |

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Citation Context ...or the solution of the inhomogeneous equation. 1. Introduction In the present paper we continue the investigation of the long--term behaviour of asymptotically autonomous evolution equations begun in =-=[30]-=-. There we studied the Cauchy problemsu(t) = A(t)u(t) + f(t); t ? ss0; u(s) = x; (1.1) on a Banach space X assuming that the linear operators A(t), ts0, are sectorial of the same type and satisfy the ... |

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Citation Context ...on 3.2 and 3.3 which show the robustness of exponential dichotomy of (parabolic) partial functional differential equations. In differing settings this subject is also treated in [4], [9], [14], [16], =-=[18]-=-, [20], [28]. In Section 4 it is then proved that the mild solution u of the inhomogeneous problem (1.5) converges to the stationary solution at infinity, i.e., lim t!1 u(t) = \Gamma(A + L 0 ) \Gamma1... |

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Citation Context ...30, Thm.4.1]. These results extend a theorem by H. Tanabe, [32, Thm.5.6.1]; see also [15] for closely related facts and [30] for further references. Very recently, C.J.K. Batty and R. Chill showed in =-=[5]-=- that one can allow for ff = 0 in (1.2) (i.e., convergence in L(X)). This paper extends [30] in several directions, e.g., the almost periodicity of U(\Delta; \Delta) is studied. We now complement (1.1... |

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Citation Context ... also employ representation formulas for the mild solution of the inhomogeneous equation investigated in Section 4. Here our methods are inspired by the evolution semigroup approach and by the papers =-=[12]-=-, [13], [14], [24]. Several auxiliary results are presented in the next section. Unexplained notation can be found in [10]. By c we denote a generic constant. 2. Prerequisites Parabolic evolution equa... |

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Citation Context ...position 3.2 and 3.3 which show the robustness of exponential dichotomy of (parabolic) partial functional differential equations. In differing settings this subject is also treated in [4], [9], [14], =-=[16]-=-, [18], [20], [28]. In Section 4 it is then proved that the mild solution u of the inhomogeneous problem (1.5) converges to the stationary solution at infinity, i.e., lim t!1 u(t) = \Gamma(A + L 0 ) \... |

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Citation Context ....3 which show the robustness of exponential dichotomy of (parabolic) partial functional differential equations. In differing settings this subject is also treated in [4], [9], [14], [16], [18], [20], =-=[28]-=-. In Section 4 it is then proved that the mild solution u of the inhomogeneous problem (1.5) converges to the stationary solution at infinity, i.e., lim t!1 u(t) = \Gamma(A + L 0 ) \Gamma1 f1 : (Here ... |

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Citation Context ...employ representation formulas for the mild solution of the inhomogeneous equation investigated in Section 4. Here our methods are inspired by the evolution semigroup approach and by the papers [12], =-=[13]-=-, [14], [24]. Several auxiliary results are presented in the next section. Unexplained notation can be found in [10]. By c we denote a generic constant. 2. Prerequisites Parabolic evolution equations.... |

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Citation Context ...on Proposition 3.2 and 3.3 which show the robustness of exponential dichotomy of (parabolic) partial functional differential equations. In differing settings this subject is also treated in [4], [9], =-=[14]-=-, [16], [18], [20], [28]. In Section 4 it is then proved that the mild solution u of the inhomogeneous problem (1.5) converges to the stationary solution at infinity, i.e., lim t!1 u(t) = \Gamma(A + L... |

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Citation Context ...convergence of solutions, variation of parameters formula, characteristic equation, evolution semigroup. 1 due to [30, Thm.4.1]. These results extend a theorem by H. Tanabe, [32, Thm.5.6.1]; see also =-=[15]-=- for closely related facts and [30] for further references. Very recently, C.J.K. Batty and R. Chill showed in [5] that one can allow for ff = 0 in (1.2) (i.e., convergence in L(X)). This paper extend... |

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Citation Context ...ation formulas for the mild solutions, whereas in Proposition 5.2 its regularity is investigated in the parabolic case. For more results in this context we refer to [9], [12], [13], [14], [23], [24], =-=[25]-=-, [31], [33, Chap.4]. Exponential dichotomy. An evolution family U(\Delta; \Delta) on X with J = [a; 1) or R is said to have an exponential dichotomy (or to be hyperbolic) if there are projections P (... |

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Citation Context ...em relies on Proposition 3.2 and 3.3 which show the robustness of exponential dichotomy of (parabolic) partial functional differential equations. In differing settings this subject is also treated in =-=[4]-=-, [9], [14], [16], [18], [20], [28]. In Section 4 it is then proved that the mild solution u of the inhomogeneous problem (1.5) converges to the stationary solution at infinity, i.e., lim t!1 u(t) = \... |

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Citation Context ...hese facts are contained in [10, xVI.6], see also [33, x3.1, 3.2] and the references therein. In this paper we extend our previous results from the undelayed case to the setting introduced above, see =-=[7]-=-, [17, x9.5], [20], [22] for related investigations if X = C n . In Theorem 3.6 we establish that the homogeneous problem (1.5) with f = 0 has an exponential dichotomy if condition (1.9) holds. This t... |

2 |
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Citation Context ...lies on Proposition 3.2 and 3.3 which show the robustness of exponential dichotomy of (parabolic) partial functional differential equations. In differing settings this subject is also treated in [4], =-=[9]-=-, [14], [16], [18], [20], [28]. In Section 4 it is then proved that the mild solution u of the inhomogeneous problem (1.5) converges to the stationary solution at infinity, i.e., lim t!1 u(t) = \Gamma... |

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Citation Context ...o small delays in the parabolic case, see [4, Thm.4.2] and [18] for related autonomous results. We point out that in general exponential stability can be destroyed by arbitrary small delays, see [4], =-=[8]-=-, [18]. The next preliminary lemma allows to prove the uniformity of the dichotomy constants needed in Proposition 3.8. Lemma 3.1. Let U(\Delta; \Delta) be an evolution family with J = R satisfying (2... |

1 |
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Citation Context ...ve representation formulas for the mild solutions, whereas in Proposition 5.2 its regularity is investigated in the parabolic case. For more results in this context we refer to [9], [12], [13], [14], =-=[23]-=-, [24], [25], [31], [33, Chap.4]. Exponential dichotomy. An evolution family U(\Delta; \Delta) on X with J = [a; 1) or R is said to have an exponential dichotomy (or to be hyperbolic) if there are pro... |

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Citation Context ...formulas for the mild solutions, whereas in Proposition 5.2 its regularity is investigated in the parabolic case. For more results in this context we refer to [9], [12], [13], [14], [23], [24], [25], =-=[31]-=-, [33, Chap.4]. Exponential dichotomy. An evolution family U(\Delta; \Delta) on X with J = [a; 1) or R is said to have an exponential dichotomy (or to be hyperbolic) if there are projections P (\Delta... |