Existence results for evolution equations with superlinear growth

Irene Benedetti, Eugénio M. Rocha

DOI: http://dx.doi.org/10.12775/TMNA.2019.101

Abstract


By combining an approximation technique with the Leray-Schauder continuation principle, we prove global existence results for semilinear differential equations involving a dissipative linear operator, generating an extendable compact $C_0$-semigroup of contractions, and a Caratheodory nonlinearity $f\colon [0,T] \times E \to F$, with $E$ and $F$ two real Banach spaces such that $E \subseteq F$, besides imposing other conditions. The case $E\neq F$ allows to treat, as an application, parabolic equations with continuous superlinear nonlinearities which satisfy a sign condition.

Keywords


Semilinear differential equation; approximation solvability method; Leray-Schauder continuation principle; Nemytskii operator

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References


J. Andres, L. Malaguti and V. Taddei, On boundary value problems in Banach spaces, Dynamic Syst. Appl. 18 (2009), 275–302.

I. Benedetti, N.V. Loi and V. Taddei, Nonlocal diffusion second order partial differential equations, Discrete Contin. Dyn. Syst. Ser. A 37 (2017), no. 6 , 2977–2998.

I. Benedetti, L. Malaguti and V. Taddei, Nonlocal solutions of parabolic equations with strongly elliptic differential operators, J. Math. Anal. Appl. 473 (2019), 421–443.

K. Deimling, Nonlinear Functional Analysis, Springer–Verlag, 1980.

G. Dinca, P. Jebelean and J. Mawhin, Variational and topological methods for Dirichlet problems with p-Laplacian, Port. Math. 58, (2001), Fasc. 3, 339–378.

M. Furi and P. Pera, A continuation method on locally convex spaces and applications to ordinary differential equations on noncompact intervals, Ann. Polon. Math. 47 (1987), 331–346.

R.E. Gaines and J.L. Mawhin, Ordinary differential equations with nonlinear boundary conditions, J. Differential Equations 26 (1977), 200–222.

V. Lakshmikantham and S. Leela, Nonlinear Differential Equations in Abstract Spaces, Pergamon Press, 1981.

J. Leray and J. Schauder, Topologie et Equations Fonctionelles, Ann. Sci. Éc. Norm. Supér. 51 (1934), 45–78.

J. Mawhin and H. B. Thompson, Periodic or bounded solutions of Carathéodory systems of ordinary differential equations, J. Dynam. Differential Equations 15 (2003), 327–334.

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer–Verlag, 1983.

M.M. Vainberg, Variational Methods for the Study of Nonlinear Operators, Holden–Day, Inc. 1964.

I.I. Vrabie, Compactness Methods for Nonlinear Evolutions, Second Edition, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 75, Longman, 1995.

I.I. Vrabie, C0 -Semigroups and Applications, North-Holland Mathematics Studies, vol. 191, North-Holland Elsevier, 2003.


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