Nonlinear perturbation of linear evolution equations in a banach space

SynopsisWe consider the ordinary differential equation [Bu(t)]′ = Au(t) with A and B linear operators with domains in a Banach space X and ranges in a Banach space Y. The initial condition is that the limit as (t → 0 of Bu(t) is prescribed in Y. We study the properties of the “solution operator” S(t) which maps the initial state in Y to the solution u(t) at time t. The notion of infinitesimal generator A of S(t) is introduced and the relationships between S(t), an associated semi-group E(t), the operators A and B and some other operators are studied. In particular a pair of operators Ao and Bo, derived from A and B, determine the family S(t) of operators. These so-called “generating pairs” are characterized. The operators A and B and Ao and Bo need not be closed, but form so-called closed pairs which is a weaker condition. We also discuss two applications of the theory.

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Integral solutions for non-linear evolution equations on Banach space

Studia Mathematica ◽

10.4064/sm-55-2-141-149 ◽

1976 ◽

Vol 55 (2) ◽

pp. 141-149

Author(s):

Nicolae Pavel

Keyword(s):

Banach Space ◽

Evolution Equations ◽

Linear Evolution ◽

Non Linear ◽

Linear Evolution Equations ◽

Integral Solutions

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3D solutions of non-linear evolution equations with diffusion

Journal de Physique ◽

10.1051/jphys:01984004503043500 ◽

1984 ◽

Vol 45 (3) ◽

pp. 435-439 ◽

Cited By ~ 21

Author(s):

H. Wilhelmsson

Keyword(s):

Evolution Equations ◽

Linear Evolution ◽

Non Linear ◽

Linear Evolution Equations

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Linear evolution equations on the half-line with dynamic boundary conditions

European Journal of Applied Mathematics ◽

10.1017/s0956792521000103 ◽

2021 ◽

pp. 1-33

Author(s):

D. A. SMITH ◽

W. Y. TOH

Keyword(s):

Boundary Conditions ◽

Heat Equation ◽

Evolution Equations ◽

Half Line ◽

Robin Problem ◽

Dynamic Boundary Conditions ◽

Transform Method ◽

Linear Evolution ◽

Robin Condition ◽

Linear Evolution Equations

The classical half-line Robin problem for the heat equation may be solved via a spatial Fourier transform method. In this work, we study the problem in which the static Robin condition $$bq(0,t) + {q_x}(0,t) = 0$$ is replaced with a dynamic Robin condition; $$b = b(t)$$ is allowed to vary in time. Applications include convective heating by a corrosive liquid. We present a solution representation and justify its validity, via an extension of the Fokas transform method. We show how to reduce the problem to a variable coefficient fractional linear ordinary differential equation for the Dirichlet boundary value. We implement the fractional Frobenius method to solve this equation and justify that the error in the approximate solution of the original problem converges appropriately. We also demonstrate an argument for existence and unicity of solutions to the original dynamic Robin problem for the heat equation. Finally, we extend these results to linear evolution equations of arbitrary spatial order on the half-line, with arbitrary linear dynamic boundary conditions.

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