Sunto
L'equazione differenziale lineare degenere in uno spazio di Banach X
viene trasformata nell'equazione multivoca du/dt +A(t) u ∋ f(t), 0 <t ⩽T. Sotto opportune ipotesi sulla norma di M(t)(zM(t)+L(t)) −1 nello spazio L(X),z numero complesso, si dimostra che −A(t)=−L(t)M(t)-− genera un semigruppo infinitamente differenziabile in X e si costruisce la soluzione fondamentale per tali problemi. I risultati vengono applicati sia a molte equazioni differenziali paraboliche degeneri nella derivata temperale, sia alla equazione di Stokes.
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References
P. Acquistapace,Evolution operators and strong solutions of abstract linear parabolic equations, Diff. Int. Eq.,1 (1988), pp. 433–457.
H. Amann,On abstract parabolic fundamental solution, J. Math. Soc. Japan,39 (1987), pp. 93–116.
G. Da Prato -P. Grisvard,Sommes d'opérateurs linéaires et équations différentielles opérationnelles, J. Math. Pures Appl.,54 (1975), pp. 305–387.
A. Favini,Laplace transform method for a class of degenerate evolution problems, Rend. Mat.,12 (1979), pp. 511–536.
A. Favini,Abstract potential operators and spectral methods for a class of degenerate evolution problems, J. Differ. Equations,39 (1981), pp. 212–225.
A. Favini,Degenerate and singular evolution equations in Banach space, Math. Ann.,273 (1985), pp. 17–44.
A. Favini,Abstract singular equations and applications, J. Math. Anal. Appl.,116 (1986), pp. 289–308.
A. Favini -P. Plazzi,On some abstract degenerate problems of parabolic type-1:the linear case, Nonlinear Anal. Theory Methods Appl.,12 (1988), pp. 1017–1027.
A. Favini -P. Plazzi,On some abstract degenerate problems of parabolic type-2:the nonlinear case, Nonlinear Anal. Theory Methods Appl.,13 (1989), pp. 23–31.
A. Favini -P. Plazzi,On some abstract degenerate problems of parabolic type-3:applications to linear and nonlinear problems, Osaka J. Math.,27(1990), pp. 321–359.
A. Favini -A. Yagi,Space and time regularity for degenerate evolution equations, J. Math. Soc. Japan,44 (1992),pp. 331–350.
A. Friedman -Z. Schuss,Degenerate evolution equations in Hilbert space, Trans. AMS,161 (1971), pp. 401–427.
T. Kato,Abstract evolution equations of parabolic type in Banach and Hilbert spaces, Nagoya Math. J.,19 (1961), pp. 93–125.
T. Kato -H. Tanabe,On the abstract evolution equation, Osaka J. Math.,14 (1962), pp. 107–133.
R. E. Showalter,A nonlinear parabolic-Sobolev equation, J. Math. Anal. Appl.,50 (1975), pp. 183–190.
P. E. Sobolevski,Parabolic equations in Banach space with an unbounded variable oper- ator, a fractional power of which has a constant domain of definition, Soviet Math. Dokl.,2 (1961), pp. 545–548.
P. E. Sobolevski,Equations of parabolic type in Banach space, Amer. Math. Soc. Transl. Ser. 2,49 (1966), pp. 1–62.
H. Tanabe,Remarks on the equations of evolution in a Banach space, Osaka Math. J.,12 (1960), pp. 145–166.
H. Tanabe,Note on singular perturbation for abstract differential equations, Osaka J. Math.,1 (1964), pp. 239- 252.
A. Yagi,On the abstract evolution equations in Banach spaces, J. Math. Soc. Japan,28 (1976), pp. 290- 303.
A. Yagi,Parabolic evolution equations in which the coefficients are the generators of infinitely differentiable semigroups, Funkc. Ekvacioj,32 (1989), pp. 107–124.
A. Yagi,Parabolic evolution equations in which the coefficients are the generators of infinitely differentiable semigroups II, Funkc. Ekvacioj,33 (1990), pp. 139–150.
A. Yagi,Generation theorem of semigroup for multivalued linear operators, Osaka J. Math.,28 (1991), pp. 385–410.
R. W. Carroll -R. E. Showalter,Singular and Degenerate Cauchy Problems, Academic Press, London, New York (1976).
S. G.Krein,Linear Differential Equations in Banach Spaces, Moskow (1967) (in Russian). English translation, Trans. Math. Mon.,29, AMS, Providence (1971).
H. Tanabe,Equations of Evolution, Iwanami, Tokyo (1975) (in Japanese). English translation, Pitman, London (1979).
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Entrato in Redazione il 25 settembre 1990.
Partially supported by Ministero della Pubblica Istruzione, Italy (Fondi 40%) and by Università di Bologna (Fondi 60%). The author is a member of G.N.A.F.A. of C.N.R.
Visiting professor in Bologna Università on C.N.R. and Università di Bologna funds.
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Favini, A., Yagi, A. Multivalued linear operators and degenerate evolution equations. Annali di Matematica pura ed applicata 163, 353–384 (1993). https://doi.org/10.1007/BF01759029
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DOI: https://doi.org/10.1007/BF01759029