Parabolic evolution equations and nonlinear boundary conditions.

*(English)*Zbl 0658.34011The author considers the initial boundary value problem
\[
(1)\quad \dot u(t)={\mathcal A}(t)u=f(t,u),\quad {\mathcal B}(t)u=g(t,u),\quad 0<t\leq T,\quad u(0)=u_ 0.
\]
\({\mathcal A}(t)\), \({\mathcal B}(t)\) are linear operators. Nonlinearities are in the right-hand side, and are present both in the main equation and in the boundary values. The goal of the author was to present an abstract theory for this problem, which allows further study of qualitative properties of the solutions, in the spirit of the geometric theory of semilinear parabolic equations by D. Henry [Lect. Notes Math. 840 (1981; Zbl 0456.35001)]. The additional fact here is the presence of nonlinear boundary conditions. An important tool for the geometric treatment is the variation of constants formula. It is precisely the purpose of this paper to derive such a formula in this “doubly” non-linear context in a rigorous way, notably regarding its domain of validity (the functional analytic framework in which it has a meaning) and its interpretation in terms of the equation (what kinds of solutions verify this formula). The paper deals in parallel with an abstract framework and its illustration by systems of second order parabolic equations. Some fundamental aspects of parabolic equations (such as analyticity of the semigroup, the availability of a “scale” of spaces) are more or less inserted as part of the abstract frame. Nevertheless, the author succeeds in his attempt to extract a number of “parabolic”-independent properties (notably, with his abstract “boundary” spaces \(\partial W\), the derivation of a “generalized” Green formula from a “formal” one, the notion of retraction in place of the Sobolev lifting property...). A key step lies in the derivation for the “linear” initial value problem [\({\mathcal B}(t)u=g(t)]\) of weak formulation in an and hoc space. Using this representation, the author proves an existence, uniqueness and continuity result for equation (1) (Theorem 12.1). Global existence results are then obtained assuming “abstract” growth conditions on f and g (Proposition 12.6), as well as a blow up result (Theorem 12.10). The paper is divided into 4 main parts: the first two are preparatory; the third one entitled “Abstract Parabolic Initial Boundary Value Problems” covers sections 8 to 12. The main existence results are in section 12. Part IV deals with applications to second order parabolic equations, culminating in theorem 15.2. This last theorem states a global existence result and precompactness of bounded orbits under “polynomial” growth conditions of f and g (conditions which, according to an example by A. Friedman and B. McLeod quoted in this paper are optimal). The interest and importance of this work are underlined by the appreciable number of contributions which have been made on specific related problems (see references there in). This is especially true in the area of population dynamics (which is the reviewer’s speciality). The only objection in that case is that the abstract framework of this paper is not large enough to include the general problems set in this domain.

Reviewer: O.Arino

##### MSC:

34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |

##### Keywords:

parabolic evolution equations; geometric theory; scale; weak solutions; maximal solutions; growth condition; semilinear parabolic equations; variation of constants formula; systems of second order parabolic equations; retraction; blow up; applications
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\textit{H. Amann}, J. Differ. Equations 72, No. 2, 201--269 (1988; Zbl 0658.34011)

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[1] | Agmon, S.; Douglis, A.; Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Comm. pure appl. math., XVII, 35-92, (1964) · Zbl 0123.28706 |

[2] | Alikakos, N.D., Regularity and asumptotic behaviour for the second order parabolic equation with nonlinear boundary conditions, J. differential equations, 39, 311-344, (1981) · Zbl 0501.35045 |

[3] | Amann, H., Gewo¨hnliche differentialgleichungen, (1983), W. de Gruyter Berlin |

[4] | Amann, H.; Amann, H., Parabolic evolution equations with nonlinear boundary conditions, (), 21-39, cf. also · Zbl 1159.35386 |

[5] | Amann, H., Existence and regularity for semilinear parabolic evolution equations, Ann. scuola norm. sup. Pisa ser. IV, XI, 593-676, (1984) · Zbl 0625.35045 |

[6] | Amann, H., Global existence for semilinear parabolic systems, J. reine angew. math., 366, 47-83, (1985) · Zbl 0564.35060 |

[7] | Amann, H., Quasilinear evolution equations and parabolic systems, Trans. amer. math. soc., 293, 191-227, (1986) · Zbl 0635.47056 |

[8] | Amann, H., Quasilinear parabolic systems under nonlinear boundary conditions, Arch. rational mech. anal., 92, 153-192, (1986) · Zbl 0596.35061 |

[9] | Amann, H., Semigroups and nonlinear evolution equations, Linear algebra appl., 84, 3-32, (1986) · Zbl 0624.35047 |

[10] | Aronson, D.G., A comparison method for stability analysis of nonlinear parabolic problems, SIAM rev., 20, 245-264, (1978) · Zbl 0384.35035 |

[11] | Balakrishnan, A.V., Applied functional analysis, (1976), Springer-Verlag New York/Berlin · Zbl 0333.93051 |

[12] | Barbu, V., Nonlinear semigroups and differential equations in Banach spaces, (1976), Nordhooff Leyden |

[13] | Bergh, J.; Loefstroem, J., Interpolation spaces: an introduction, (1976), Springer-Verlag Berlin/New York |

[14] | Bhatia, N.P.; Szegoe, G.P., Dynamical systems: stability theory and applications, () |

[15] | Bre´zis, H., Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations, (), 101-156 |

[16] | Degiovanni, M., Parabolic equations with nonlinear time-dependent boundary conditions, Ann. mat. pura appl., 141, 223-263, (1985) · Zbl 0592.35070 |

[17] | Desch, W.; Lasiecka, I.; Schappacher, W., Feedback boundary control problems for linear semigroups, Israel J. math., 51, 177-207, (1985) · Zbl 0587.93038 |

[18] | Dubois, R.M.; Lumer, G., Formule de Duhamel abstraite, Arch. math., 43, 49-56, (1984) · Zbl 0555.35058 |

[19] | Friedman, A., Generalized heat transfer between solids and gases under nonlinear boundary conditions, J. math. mech., 8, 161-184, (1959) · Zbl 0101.31102 |

[20] | Friedman, A., Partial differential equations of parabolic type, (1964), Prentice-Hall Englewood Cliffs, NY · Zbl 0144.34903 |

[21] | Friedman, A.; McLeod, B., Blow-up of positive solutions of semilinear heat equations, Indiana univ. math. J., 34, 425-447, (1985) · Zbl 0576.35068 |

[22] | Geymonat, G., Su alcuni problemi ai limiti per i sistemi lineari ellittici secondo petrowsky, Le matematiche, 20, 211-253, (1965) · Zbl 0143.14401 |

[23] | \scM. Giaquinta and G. Modica, Local existence for quasilinear parabolic systems under nonlinear boundary conditions, preprint. · Zbl 0655.35049 |

[24] | \scJ. A. Goldstein, Evolution equations with nonlinear boundary conditions, preprint. · Zbl 0723.35037 |

[25] | \scG. Greiner, Perturbing the boundary conditions of a generator,Houston J. Math., in press. · Zbl 0639.47034 |

[26] | Grisvard, P., E´quations diffe´rentielles abstraites, Ann. sci.E´cole norm. sup. (4), 2, 311-395, (1969) · Zbl 0193.43502 |

[27] | Grisvard, P., Elliptic problems in nonsmooth domains, (1985), Pitman Boston · Zbl 0695.35060 |

[28] | Grubb, G., Weakly semibounded boundary problems and sesquilinear forms, Ann. inst. Fourier, 23, 4, 145-194, (1973) · Zbl 0261.35011 |

[29] | Henry, D., Geometric theory of semilinear parabolic equations, () · Zbl 0456.35001 |

[30] | Henry, D., Some infinite-dimensional Morse-Smale systems defined by parabolic partial differential equations, J. differential equations, 59, 165-205, (1985) · Zbl 0572.58012 |

[31] | Hernandez, J., Some existence and stability results for solutions of reaction-diffusion systems with nonlinear boundary conditions, (), 161-173 |

[32] | Hoermander, L., Definitions of maximal differential operators, Ark. mat., 3, 500-504, (1958) |

[33] | Kellermann, Linear evolution equations with time-dependent domain, Universita¨t tu¨bingen: semesterbericht funktional-analysis, 9, 15-44, (1985/1986) |

[34] | Ladyzˇenskaja, O.A.; Solonnikov, V.A.; Ural’ceva, N.N., Linear and quasilinear-equations of parabolic type, (1968), Amer. Math. Soc. Providence, RI |

[35] | Lasiecka, I.; Triggiani, R., Stabilization of neuman boundary feedback of parabolic equations. the case of trace in the feedback loop, Appl. math. optim., 10, 307-350, (1983) · Zbl 0524.93052 |

[36] | Leung, A., A semilinear reaction-diffusion prey-predator system with nonlinear coupled boundary conditions: equilibrium and stability, Indiana univ. math. J., 31, 223-241, (1982) · Zbl 0519.92021 |

[37] | Levine, H.A., Some nonexistence and instability theorems for formally parabolic equations of the form pu_t = − au + F(u), Arch. rational mech. anal., 51, 371-386, (1973) · Zbl 0278.35052 |

[38] | Lions, J.-L.; Magenes, E., Problemi ai limiti non omogenei (V), Ann. scuola norm. sup. Pisa, XVI, 1-44, (1962) · Zbl 0115.31401 |

[39] | Lions, J.-L.; Magenes, E., Non-homogeneous boundary value problems and applications I, (1972), Springer-Verlag Berlin/New York · Zbl 0223.35039 |

[40] | Morrey, Ch.B., Multiple integrals in the calculus of variations, (1966), Springer-Verlag New York/Berlin |

[41] | Mueller, C.E.; Weissler, F.B., Single point blow-up for a general semilinear heat equation, Indiana univ. math. J., 34, 881-913, (1985) · Zbl 0597.35057 |

[42] | () |

[43] | Pao, C.V., Reaction diffusion equations with nonlinear boundary conditions, Nonlinear anal. TM & A, 5, 1077-1094, (1981) · Zbl 0519.35038 |

[44] | Seeley, R., Interpolation in L^p with boundary conditions, Stud. math., XLIV, 47-60, (1972) · Zbl 0237.46041 |

[45] | Sobolevskii, P.E., Equations of parabolic type in a Banach space, Amer. math. soc. transl. ser. 2, 49, 1-62, (1966) · Zbl 0178.50301 |

[46] | Tanabe, H., In the equation of evolution in a Banach space, Osaka math. J., 12, 363-376, (1960) · Zbl 0098.31301 |

[47] | Tanabe, H., Equations of evolution, (1979), Pitman London |

[48] | Thames, H.D.; Aronson, D.G., Oscillation in a nonlinear parabolic model of separated, cooperatively coupled enzymes, (), 687-693 |

[49] | Triebel, H., Interpolation theory, function spaces, differential operators, (1978), North-Holland Amsterdam · Zbl 0387.46032 |

[50] | Yosida, K., Functional analysis, (1965), Springer-Verlag Berlin/New York · Zbl 0126.11504 |

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