Fractional abstract Cauchy problem on complex interpolation scales

2020 ◽  
Vol 23 (4) ◽  
pp. 1125-1140
Author(s):  
Andriy Lopushansky ◽  
Oleh Lopushansky ◽  
Anna Szpila
Keyword(s):  
Cauchy Problem ◽  
Time Derivative ◽  
Abstract Cauchy Problem ◽  
Initial Boundary ◽  
Sectorial Operator ◽  
Complex Interpolation ◽  
Bounded Domains ◽  
Initial Boundary Value Problems ◽  
Fractional Time Derivative ◽  
Initial Boundary Value

AbstractAn fractional abstract Cauchy problem generated by a sectorial operator is investigated. An inequality of coercivity type for its solution with respect to a complex interpolation scale generated by a sectorial operator with the same parameters is established. An application to differential parabolic initial-boundary value problems in bounded domains with a fractional time derivative is shown.

On the time-asymptotic behaviour of solutions in thermoelasticity

1987 ◽  
Vol 107 (3-4) ◽  
pp. 289-298 ◽  
Author(s):  
Reinhard Racke
Keyword(s):  
Boundary Value ◽  
Unbounded Domains ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Bounded Domains ◽  
Initial Boundary Value Problems ◽  
Time Harmonic ◽  
Linear Thermoelasticity ◽  
Initial Boundary Value ◽  
Natural Decomposition

SynopsisWe consider initial boundary value problems for the equations of linear thermoelasticity in both bounded and unbounded domains and for both nonhomogeneous and anisotropic media. For bounded domains, it is shown that the unique solution of the problem is time-asymptotically equal to the solution of a particular initial boundary value problem which is obtained from a natural decomposition of the original initial data and which represents a (in general non-vanishing) time harmonic part. For the unbounded case similar results are obtained, but now in the sense of weak convergence which lead to the result of local energy decay: the solution tends to zero in every compactum.

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