Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems.

*(English)*Zbl 0810.35037
Schmeisser, Hans-Jürgen (ed.) et al., Function spaces, differential operators and nonlinear analysis. Survey articles and communications of the international conference held in Friedrichsroda, Germany, September 20-26, 1992. Stuttgart: B. G. Teubner Verlagsgesellschaft. Teubner-Texte Math. 133, 9-126 (1993).

Summary: It is the purpose of this paper to describe some of the recent developments in the mathematical theory of linear and quasilinear elliptic and parabolic systems with nonhomogeneous boundary conditions. For illustration we use the relatively simple set-up of reaction- diffusion systems which are – on the one hand – typically for the whole class of systems to which the general theory applies and – on the other hand – still simple enough to be easily described without too many technicalities. In addition, quasilinear reaction-diffusion equations are of great importance in applications and of actual mathematical and physical interest, as is witnessed by the examples we include.

In particular, we try to elucidate the rôles which are played in the theory of quasilinear parabolic systems by the modern theory of function spaces, interpolation and extrapolation techniques, and semigroup theory. Many of the proofs will be sketched only and we will be rather brief at times. However, we try to explain the basic underlying ideas and give references to the research literature where proofs can be found. A complete, detailed, and coherent presentation will be given in a forthcoming monograph which will also contain many additional results and extensions of the theory described in this paper.

For the entire collection see [Zbl 0782.00088].

In particular, we try to elucidate the rôles which are played in the theory of quasilinear parabolic systems by the modern theory of function spaces, interpolation and extrapolation techniques, and semigroup theory. Many of the proofs will be sketched only and we will be rather brief at times. However, we try to explain the basic underlying ideas and give references to the research literature where proofs can be found. A complete, detailed, and coherent presentation will be given in a forthcoming monograph which will also contain many additional results and extensions of the theory described in this paper.

For the entire collection see [Zbl 0782.00088].

##### MSC:

35K60 | Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations |

35J65 | Nonlinear boundary value problems for linear elliptic equations |

35K50 | Systems of parabolic equations, boundary value problems (MSC2000) |

35J55 | Systems of elliptic equations, boundary value problems (MSC2000) |

47H20 | Semigroups of nonlinear operators |

47D06 | One-parameter semigroups and linear evolution equations |

##### Keywords:

nonhomogeneous boundary conditions; quasilinear parabolic systems; interpolation; extrapolation
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\textit{H. Amann}, in: Function spaces, differential operators and nonlinear analysis. Survey articles and communications of the international conference held in Friedrichsroda, Germany, September 20-26, 1992. Stuttgart: B. G. Teubner Verlagsgesellschaft. 9--126 (1993; Zbl 0810.35037)