We study the stability of the null solution of a class of nonlinear evolution equations in Banach space. After stating a local existence result and the principle of linearized stability, we study the critical case, giving sufficient conditions for stability. The results are applied to second-order fully nonlinear parabolic equations in [0, + ∞ [ × R n.
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S. Angenent, Nonlinear Analytic Semi-flows, Proc. Royal Soc. Edinb. 115A (1990), 91–107.
P. Cannarsa & V. Vespri, Generation of analytic semigroups by elliptic operators with unbounded coefficients, SIAM J. Math. Anal. 18 (1987), 857–872.
G. Da Prato & P. Grisvard, Equations d'évolution abstraites non linéaires de type parabolique, Ann Mat. Pura Appl. (IV) 120 (1979), 329–396.
G. Da Prato & A. Lunardi, Stability, instability and center manifold theorem for fully nonlinear autonomous parabolic equations in Banach space, Arch. Rational Mech. Anal. 101 (1988), 115–141.
A. K. Drangeid, The principle of linearized stability for quasilinear parabolic evolution equations, Nonlinear Analysis T.M.A. 13 (1989), 1091–1113.
H. Fujita, On the blowing up of solutions of the Cauchy problem for ut=Δu + u 1+α, J. Fac. Sci. Univ. Tokyo, Sect. 1, 13 (1966), 109–124.
D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Math. 840, Springer-Verlag, New York (1981).
D. Gilbarg & N. S. Trudinger, Elliptic partial differential equations of second order, 2nd Edition, Springer-Verlag, Berlin (1983).
S. Klainerman, Long-time behavior of solutions to nonlinear evolution equations, Arch. Rational Mech. Anal. 78 (1982), 73–98.
N. V. Krylov, Nonlinear Elliptic and Parabolic Equations of the Second Order, Nauka, Moscow (1985), English transl.: D. Reidel Publishing Co., Dordrecht (1987).
A. Lunardi, Asymptotic exponential stability in quasilinear parabolic equations, Nonlinear Analysis T.M.A. 9 (1985), 563–586.
A. Lunardi, On the local dynamical system associated to a fully nonlinear parabolic equation, in: Nonlinear Analysis and Applications, V. Lakshmikantham, Ed., Marcel Dekker Publ., 1987, 319–326.
A. Lunardi, On the evolution operator for abstract parabolic equations, Israel J. Math. 60 (1987), 281–314.
A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser Verlag, Basel (to appear).
A. Lunardi & E. Sinestrari, Fully nonlinear integrodifferential equations in general Banach space, Math. Z. 190 (1985), 225–248.
A. Lunardi & E. Sinestrari, Existence in the large and stability for nonlinear Volterra equations, in: “Integrodifferential evolution equations and applications”, J. Int. Eq. 10 (1985), Special Conference Issue, 213–239.
M. Miklavčič, Stability for semilinear parabolic equations with noninvertible linear operator, Pacific J. Math. 118 (1985), 199–214.
M. Potier-Ferry, The linearization principle for the stability of solutions of quasilinear parabolic equations, I, Arch. Rational Mech. Anal. 77 (1981), 301–320.
G. Ponce, Global existence of small solutions to a class of nonlinear evolution equations, Nonlinear Analysis T.M.A. 9 (1985), 399–418.
M. Reed & B. Simon, Methods of modern mathematical physics. II: Fourier Analysis, self-adjointness, Academic Press, New York (1975).
E. Sinestrari, On the abstract Cauchy problem of parabolic type in spaces of continuous functions, J. Math. Anal. Appl. 107 (1985), 16–66.
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam (1978).
F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Isr. J. Math. 38 (1981), 29–40.
S. Zheng & Y. Chen, Global existence for nonlinear equations, Chinese Ann. Math. 6B (1986), 57–73.
S. Zheng, Remarks on global existence for nonlinear parabolic equations, Nonlinear Analysis T.M.A. 10 (1986), 107–114.
Communicated by C. Dafermos
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Lunardi, A. Stability in fully nonlinear parabolic equations. Arch. Rational Mech. Anal. 130, 1–24 (1995). https://doi.org/10.1007/BF00375654
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