Irregular convergence of mild solutions of semilinear equations

Bobrowski, Adam; Kunze, Markus

doi:10.1016/j.jmaa.2018.11.082

Cited by 11 publications

(15 citation statements)

References 28 publications

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“…4 and the proof of Proposition 8.2 in [24]). Furthermore, by the main result of [22], solutions to semilinear problems with a globally Lipchitz nonlinear term converge (even in a degenerate manner) iff so do the solutions to the corresponding linear problems.…”

Section: Nonlinearitymentioning

confidence: 99%

“…From the perspective of the latter result, it may seem that including nonlinear terms does not add value to the paper. Nevertheless, because of the motivations outlined above, we feel that disposing of these terms would make the results less natural and interesting (and the paper [22] is not so well known).…”

Section: Nonlinearitymentioning

confidence: 99%

“…This is a system of reaction-diffusion equations involving two-dimensional diffusion on the two sides of the membrane B. As in [22][23][24], in the limit the boundary conditions (2.2) 'jump into' the main equation to form new terms of the reaction part. If c − and c + are non-negative, the terms −c − u − and − c + u + describe the process of random 'killing' of diffusing particles on the upper and lower sides, respectively.…”

Section: Intuitions and The Limit Master Equationmentioning

confidence: 99%

“…where F on the right-hand side is the function from (2.1). Assuming that F(0) = 0 or, more generally, that there is a u ∈ L 2 (Ω) such that F(u) ∈ L 2 (Ω), we check, using the existence of a global Lipschitz constant for F, that (2.9) defines a globally Lipschitz continuous map L 2 (Ω) → L 2 (Ω), with the Lipschitz constant inherited from F. In (2.8), for simplicity of notation, we do not distinguish between F and F. As discussed in [22] and our Sect. 1.5, the problems of well-posedness and convergence of solutions to (2.8), reduce to these for the related problem without the nonlinear term (see also the end of Sect.…”

Section: A View Through a Magnifying Glassmentioning

confidence: 99%

“…Hence, it remains to take care of the nonlinearity. By the main theorem of [22], however, convergence of semigroups e t A ε t≥0 , even in a degenerate manner, as in (2.17), implies convergence of mild solutions of (2.1)-(2.3) to solutions of ∂ t u(t) = Bu(t) + P F(u(t)).…”

Section: Convergence As ε →mentioning

confidence: 99%

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Semigroup-theoretic approach to diffusion in thin layers separated by semi-permeable membranes

Bobrowski

2020

J. Evol. Equ.

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Using techniques of the theory of semigroups of linear operators, we study the question of approximating solutions to equations governing diffusion in thin layers separated by a semi-permeable membrane. We show that as thickness of the layers converges to 0, the solutions, which by nature are functions of 3 variables, gradually lose dependence on the vertical variable and thus may be regarded as functions of 2 variables. The limit equation describes diffusion on the lower and upper sides of a two-dimensional surface (the membrane) with jumps from one side to the other. The latter possibility is expressed as an additional term in the generator of the limit semigroup, and this term is built from permeability coefficients of the membrane featuring in the transmission conditions of the approximating equations (i.e., in the description of the domains of the generators of the approximating semigroups). We prove this convergence result in the spaces of square integrable and continuous functions, and study the way the choice of transmission conditions influences the limit.

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Section: Nonlinearitymentioning

confidence: 99%

Section: Nonlinearitymentioning

confidence: 99%

Section: Intuitions and The Limit Master Equationmentioning

confidence: 99%

Section: A View Through a Magnifying Glassmentioning

confidence: 99%

Section: Convergence As ε →mentioning

confidence: 99%

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Semigroup-theoretic approach to diffusion in thin layers separated by semi-permeable membranes

Bobrowski

2020

J. Evol. Equ.

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Dynamics of Shadow System of a Singular Gierer–Meinhardt System on an Evolving Domain

2020

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The main purpose of the current paper is to contribute towards the comprehension of the dynamics of the shadow system of a singular Gierer–Meinhardt model on an isotropically evolving domain. In the case where the inhibitor’s response to the activator’s growth is rather weak, then the shadow system of the Gierer–Meinhardt model is reduced to a single though non-local equation whose dynamics is thoroughly investigated throughout the manuscript. The main focus is on the derivation of blow-up results for this non-local equation, which can be interpreted as instability patterns of the shadow system. In particular, a diffusion-driven instability (DDI), or Turing instability, in the neighbourhood of a constant stationary solution, which then is destabilised via diffusion-driven blow-up, is observed. The latter indicates the formation of some unstable patterns, whilst some stability results of global-in-time solutions towards non-constant steady states guarantee the occurrence of some stable patterns. Most of the theoretical results are verified numerically, whilst the numerical approach is also used to exhibit the dynamics of the shadow system when analytical methods fail.

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