## Existence and uniqueness for a degenerate parabolic equation with L 1 -data (1999)

Venue: | 285 306. VARIATION FLOW |

Citations: | 14 - 4 self |

### BibTeX

@INPROCEEDINGS{Andreu99existenceand,

author = {F. Andreu and J. M. Mazón and S. Segura and De León and J. Toledo},

title = {Existence and uniqueness for a degenerate parabolic equation with L 1 -data},

booktitle = {285 306. VARIATION FLOW},

year = {1999}

}

### OpenURL

### Abstract

Abstract. In this paper we study existence and uniqueness of solutions for the boundary-value problem, with initial datum in L1 (Ω), ut =diva(x, Du) in (0,∞)×Ω, − ∂u ∈ β(u)

### Citations

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Citation Context ... American Mathematical Society286 F. ANDREU, J. M. MAZÓN, S. SEGURA DE LEÓN, AND J. TOLEDO The hypotheses (H1), (H2) and(H3) are classical in the study of nonlinear operators in divergence form (see =-=[L]-=-). The model example of a function a satisfying these hypothesis is a(x, ξ) =|ξ| p−2ξ. The corresponding operator is the p-Laplacian operator ∆p(u) =div ( |Du| p−2 Du ) . This operator has been widely... |

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Citation Context ... ) function Tku pointwise, i.e., for every x ∈ Ω the value of Tku at x is just Tk u(x) .Observethat 1 lim k→0 k Tk(s) ⎧ ⎪⎨1 if s>0, =sign(s):= 0 if s =0, ⎪⎩ −1 if s<0. By the Stampacchia Theorem, cf. =-=[KS]-=-, if u ∈ W 1,1 (Ω), we have DTk(u) =1 {|u|<k}Du, where 1B denotes the characteristic function of a measurable set B ⊂ Ω. We need to define the trace of functions which are not in the Sobolev spaces. B... |

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Citation Context ...1,1 loc (Ω) is defined as the unique function v satisfying (2.1). This notation will be used throughout in the sequel. Let Ω be a bounded open subset of RN of class C1 ,and1≤p<∞. It is wellknown (cf. =-=[N]-=- or [M]) that if u ∈ W 1,p (Ω), it is possible to define the trace of u on ∂Ω. More precisely, there exists a bounded operator γ from W 1,p (Ω) into Lp (∂Ω) such that γ(u) =u|∂Ωwhenever u ∈ C(Ω). Now,... |

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Citation Context ...dary conditions see [X]. With respect to existence-uniqueness results for Cauchy problems of type (E), we only know the one given by E. Di Benedetto and M. A. Herrero in [DiH-1] and [DiH-2] (see also =-=[Di-1]-=- and [Di-2]) for the p-Laplacian equation in RN . In this case, they introduce aDEGENERATE PARABOLIC EQUATION 287 class of weak solutions and prove existence and uniqueness of this type of solutions ... |

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Citation Context ...n L p (QT ). Now, since Tkun → Tk(u) in L p (QT), it follows that DTkun → DTk(u) weaklyin L p (QT). We now prove that {Dun}n∈N isaCauchysequenceinmeasure. Todothiswe follow the same technique used in =-=[BG-1]-=- (see also [AMST]). Let r, ɛ > 0. For some A>1, we set C(x, A, r) :=inf{〈a(x, ξ) − a(x, η),ξ−η〉 : |ξ|≤A, |η| ≤A, |ξ − η| ≥r}. Having in mind that the function ψ → a(x, ψ) is continuous for almost all ... |

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Citation Context ...h that ζn → Tku in L 1 loc (Ω), Dζn → DTk(u) in L p (Ω). As a consequence of the characterizations of T 1,p 0 (Ω) given in [B-V, Appendix II] we have Ker(τ) =T 1,p 0 (Ω). We refer the reader to [Ba], =-=[Be]-=-, [BCP] and [Cr] for background material on the theory of nonlinear semigroups. 3. The case of bounded Ω Throughout this section Ω is a bounded domain in RN (N ≥ 2) with smooth boundary ∂Ω ofclassC1 ,... |

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Citation Context ...pe (E) has received a great deal of attention. For example, existence of weak solutions with measures as initial data, in the case of Ω bounded with Dirichlet boundary conditions, has been studied in =-=[BG-2]-=-, [Ra-1] and [Ra-2]. For some results about existence of weak solutions of similar equations with non-linear boundary conditions see [X]. With respect to existence-uniqueness results for Cauchy proble... |

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Citation Context ...operators for which the classical theory is not available. It also appears in several physical problems—for instance, in non-Newtonian fluids (see [DH] and the literature cited therein). Recently, in =-=[B-V]-=-, a new concept of solution has been introduced for the elliptic equation −div a(x, Du) =f(x) in Ω, u =0 on∂Ω, namely entropy solution. As a consequence, an m-completely accretive operator in L1 (Ω) c... |

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Citation Context ...ity of the initial data and the homogeneity of the p-Laplacian are essential in their proof of uniqueness, since they use some sort of time-compactness via the regularizing effect of Bénilan-Crandall =-=[BCr-1]-=-. Di Benedetto in [Di-2] says the following: “It would be of interest to have a notion of solution that is irrespective of the sign of the solution and a correspondent existence-uniqueness theorem. We... |

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Citation Context ...in L 1 loc (Ω), Dζn → DTk(u) in L p (Ω). As a consequence of the characterizations of T 1,p 0 (Ω) given in [B-V, Appendix II] we have Ker(τ) =T 1,p 0 (Ω). We refer the reader to [Ba], [Be], [BCP] and =-=[Cr]-=- for background material on the theory of nonlinear semigroups. 3. The case of bounded Ω Throughout this section Ω is a bounded domain in RN (N ≥ 2) with smooth boundary ∂Ω ofclassC1 ,1<p<N,ais a vect... |

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Citation Context ...equations with non-linear boundary conditions see [X]. With respect to existence-uniqueness results for Cauchy problems of type (E), we only know the one given by E. Di Benedetto and M. A. Herrero in =-=[DiH-1]-=- and [DiH-2] (see also [Di-1] and [Di-2]) for the p-Laplacian equation in RN . In this case, they introduce aDEGENERATE PARABOLIC EQUATION 287 class of weak solutions and prove existence and uniquene... |

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Citation Context ...n by the exponential formula ( t S(t)u0 = lim I + n→∞ n A) −n u0. Moreover, since A is completely accretive, if the initial datum u0 ∈D(A) then the mild solution u(t) =S(t)u0is a strong solution (see =-=[BCr-2]-=-), i.e., u ∈ W 1,1 (0,T;L 1 (Ω)) and (II) is satisfied almost everywhere. The next result is a consequence of nonlinear semigroup theory. We include the proof here for the sake of completeness. Lemma ... |

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Citation Context ... ζn → Tku in L 1 loc (Ω), Dζn → DTk(u) in L p (Ω). As a consequence of the characterizations of T 1,p 0 (Ω) given in [B-V, Appendix II] we have Ker(τ) =T 1,p 0 (Ω). We refer the reader to [Ba], [Be], =-=[BCP]-=- and [Cr] for background material on the theory of nonlinear semigroups. 3. The case of bounded Ω Throughout this section Ω is a bounded domain in RN (N ≥ 2) with smooth boundary ∂Ω ofclassC1 ,1<p<N,a... |

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Citation Context ...tic equation −div a(x, Du) =f(x) in Ω, u =0 on∂Ω, namely entropy solution. As a consequence, an m-completely accretive operator in L1 (Ω) can be associated to the corresponding parabolic equation. In =-=[AMST]-=-, using the method developed in [B-V], we study entropy solutions for the elliptic problem with non-linear boundary conditions. Precisely, we study existence and uniqueness of entropy solutions for eq... |

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Citation Context ... one of the simpler examples of degenerate nonlinear operators for which the classical theory is not available. It also appears in several physical problems—for instance, in non-Newtonian fluids (see =-=[DH]-=- and the literature cited therein). Recently, in [B-V], a new concept of solution has been introduced for the elliptic equation −div a(x, Du) =f(x) in Ω, u =0 on∂Ω, namely entropy solution. As a conse... |

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Citation Context ... a great deal of attention. For example, existence of weak solutions with measures as initial data, in the case of Ω bounded with Dirichlet boundary conditions, has been studied in [BG-2], [Ra-1] and =-=[Ra-2]-=-. For some results about existence of weak solutions of similar equations with non-linear boundary conditions see [X]. With respect to existence-uniqueness results for Cauchy problems of type (E), we ... |

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Citation Context ...as received a great deal of attention. For example, existence of weak solutions with measures as initial data, in the case of Ω bounded with Dirichlet boundary conditions, has been studied in [BG-2], =-=[Ra-1]-=- and [Ra-2]. For some results about existence of weak solutions of similar equations with non-linear boundary conditions see [X]. With respect to existence-uniqueness results for Cauchy problems of ty... |

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Citation Context ...ions see [X]. With respect to existence-uniqueness results for Cauchy problems of type (E), we only know the one given by E. Di Benedetto and M. A. Herrero in [DiH-1] and [DiH-2] (see also [Di-1] and =-=[Di-2]-=-) for the p-Laplacian equation in RN . In this case, they introduce aDEGENERATE PARABOLIC EQUATION 287 class of weak solutions and prove existence and uniqueness of this type of solutions when the in... |

1 |
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Citation Context ... the unit outward normal on ∂Ω, Du the gradient of u and β a maximal monotone graph in R × R with 0 ∈ β(0). These nonlinear fluxes on the boundary occur in heat transfer between solids and gases (cf. =-=[Fr]-=-) and in some problems − ∂u ∂ηa in mechanics and physics [DL] (see also [Br-2]). Observe also that the classical Neumann and Dirichlet boundary conditions correspond to β = R ×{0}and β = {0}×R, respec... |

1 |
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Citation Context ...nded with Dirichlet boundary conditions, has been studied in [BG-2], [Ra-1] and [Ra-2]. For some results about existence of weak solutions of similar equations with non-linear boundary conditions see =-=[X]-=-. With respect to existence-uniqueness results for Cauchy problems of type (E), we only know the one given by E. Di Benedetto and M. A. Herrero in [DiH-1] and [DiH-2] (see also [Di-1] and [Di-2]) for ... |