scholarly journals On the principle of linearized stability in interpolation spaces for quasilinear evolution equations

2019 ◽  
Vol 191 (3) ◽  
pp. 615-634 ◽  
Author(s):  
Bogdan-Vasile Matioc ◽  
Christoph Walker
Keyword(s):  
Evolution Equations ◽  
Interpolation Spaces ◽  
Linearized Stability

scholarly journals Existence of Solutions in Some Interpolation Spaces for a Class of Semilinear Evolution Equations with Nonlocal Initial Conditions

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Jung-Chan Chang ◽  
Hsiang Liu
Keyword(s):  
Evolution Equations ◽  
Existence Of Solutions ◽  
Initial Conditions ◽  
Linear Part ◽  
Nonlocal Conditions ◽  
Strong Solutions ◽  
Interpolation Spaces ◽  
Nonlocal Initial Conditions ◽  
Densely Defined ◽  
Semilinear Evolution Equations

This paper is concerned with the existence of mild and strong solutions for a class of semilinear evolution equations with nonlocal initial conditions. The linear part is assumed to be a (not necessarily densely defined) sectorial operator in a Banach spaceX. Considering the equations in the norm of some interpolation spaces betweenXand the domain of the linear part, we generalize the recent conclusions on this topic. The obtained results will be applied to a class of semilinear functional partial differential equations with nonlocal conditions.

scholarly journals On the Behaviour of Singular Semigroups in Intermediate and Interpolation Spaces and Its Applications to Maximal Regularity for Degenerate Integro-Differential Evolution Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-37 ◽  
Author(s):  
Alberto Favaron ◽  
Angelo Favini
Keyword(s):  
Differential Evolution ◽  
Banach Spaces ◽  
Wide Class ◽  
Evolution Equations ◽  
Maximal Regularity ◽  
Linear Operators ◽  
Interpolation Spaces ◽  
Power Type ◽  
Maximal Time

For those semigroups, which may have power type singularities and whose generators are abstract multivalued linear operators, we characterize the behaviour with respect to a certain set of intermediate and interpolation spaces. The obtained results are then applied to provide maximal time regularity for the solutions to a wide class of degenerate integro- and non-integro-differential evolution equations in Banach spaces.

scholarly journals The periodic version of the Da Prato–Grisvard theorem and applications to the bidomain equations with FitzHugh–Nagumo transport

2020 ◽  
Vol 199 (6) ◽  
pp. 2435-2457
Author(s):  
Matthias Hieber ◽  
Naoto Kajiwara ◽  
Klaus Kress ◽  
Patrick Tolksdorf
Keyword(s):  
Periodic Solution ◽  
Evolution Equations ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
Ionic Transport ◽  
Real Interpolation ◽  
Interpolation Spaces ◽  
Bidomain Equations ◽  
Periodic Version ◽  
Semilinear Evolution Equations

Abstract In this article, the periodic version of the classical Da Prato–Grisvard theorem on maximal $${{L}}^p$$ L p -regularity in real interpolation spaces is developed, as well as its extension to semilinear evolution equations. Applying this technique to the bidomain equations subject to ionic transport described by the models of FitzHugh–Nagumo, Aliev–Panfilov, or Rogers–McCulloch, it is proved that this set of equations admits a unique, strongT-periodic solution in a neighborhood of stable equilibrium points provided it is innervated by T-periodic forces.

scholarly journals Quasilinear Evolution Equations in LμP-Spaces with Lower Regular Initial Data

2018 ◽  
Vol 2018 ◽  
pp. 1-11
Author(s):  
Qinghua Zhang
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Parabolic Equations ◽  
Evolution Equations ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Interpolation Spaces ◽  
Navier Stokes Equations ◽  
Sectorial Operators ◽  
The Cauchy Problem

We study the Cauchy problem of the quasilinear evolution equations in Lμp-spaces. Based on the theories of maximal Lp-regularity of sectorial operators, interpolation spaces, and time-weighted Lp-spaces, we establish the local posedness for a class of abstract quasilinear evolution equations with lower regular initial data. To illustrate our results, we also deal with the second-order parabolic equations and the Navier-Stokes equations in Lp,q-spaces with temporal weights.

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