### Abstract

The theories developed in the previous chapters can be used to study some parabolic equations as well. In this chapter, we first consider linear abstract Cauchy problems with non-densely defined and almost sectorial operators; that is, the part of this operator in the closure of its domain is sectorial. Such problems naturally arise for parabolic equations with nonhomogeneous boundary conditions. By using the integrated semigroup theory, we prove an existence and uniqueness result for integrated solutions. Moreover, we study the linear perturbation problem. Then in the second section we provide detailed stability and bifurcation analyses for a scalar reaction-diffusion equation, namely, a size-structured model.

Original language | English (US) |
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Title of host publication | Applied Mathematical Sciences (Switzerland) |

Publisher | Springer |

Pages | 451-521 |

Number of pages | 71 |

DOIs | |

State | Published - Jan 1 2018 |

### Publication series

Name | Applied Mathematical Sciences (Switzerland) |
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Volume | 201 |

ISSN (Print) | 0066-5452 |

ISSN (Electronic) | 2196-968X |

### Fingerprint

### ASJC Scopus subject areas

- Applied Mathematics

### Cite this

*Applied Mathematical Sciences (Switzerland)*(pp. 451-521). (Applied Mathematical Sciences (Switzerland); Vol. 201). Springer. https://doi.org/10.1007/978-3-030-01506-0_9

**Parabolic Equations.** / Magal, Pierre; Ruan, Shigui.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

*Applied Mathematical Sciences (Switzerland).*Applied Mathematical Sciences (Switzerland), vol. 201, Springer, pp. 451-521. https://doi.org/10.1007/978-3-030-01506-0_9

}

TY - CHAP

T1 - Parabolic Equations

AU - Magal, Pierre

AU - Ruan, Shigui

PY - 2018/1/1

Y1 - 2018/1/1

N2 - The theories developed in the previous chapters can be used to study some parabolic equations as well. In this chapter, we first consider linear abstract Cauchy problems with non-densely defined and almost sectorial operators; that is, the part of this operator in the closure of its domain is sectorial. Such problems naturally arise for parabolic equations with nonhomogeneous boundary conditions. By using the integrated semigroup theory, we prove an existence and uniqueness result for integrated solutions. Moreover, we study the linear perturbation problem. Then in the second section we provide detailed stability and bifurcation analyses for a scalar reaction-diffusion equation, namely, a size-structured model.

AB - The theories developed in the previous chapters can be used to study some parabolic equations as well. In this chapter, we first consider linear abstract Cauchy problems with non-densely defined and almost sectorial operators; that is, the part of this operator in the closure of its domain is sectorial. Such problems naturally arise for parabolic equations with nonhomogeneous boundary conditions. By using the integrated semigroup theory, we prove an existence and uniqueness result for integrated solutions. Moreover, we study the linear perturbation problem. Then in the second section we provide detailed stability and bifurcation analyses for a scalar reaction-diffusion equation, namely, a size-structured model.

UR - http://www.scopus.com/inward/record.url?scp=85068162991&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85068162991&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-01506-0_9

DO - 10.1007/978-3-030-01506-0_9

M3 - Chapter

T3 - Applied Mathematical Sciences (Switzerland)

SP - 451

EP - 521

BT - Applied Mathematical Sciences (Switzerland)

PB - Springer

ER -