Trotter’s product formula for semigroups generated by quasilinear elliptic operators
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- by Michiaki Watanabe
- Proc. Amer. Math. Soc. 92 (1984), 509-514
- DOI: https://doi.org/10.1090/S0002-9939-1984-0760935-8
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Abstract:
Trotter’s product formula is given for nonlinear semigroups in ${L^1}({R^N})$ generated by quasilinear operators of the form $\Delta \phi$, where $\phi$ is a suitable function: formally $\exp (t\Delta \phi )u = {\lim _{h \downarrow 0}}{\left \{ {\exp (h\Delta {\phi _1}) \cdots \exp (h\Delta {\phi _k})} \right \}^{[t/h]}}u$, where $\phi = {\phi _1} + \cdots + {\phi _k}$. The proof is carried out by a new method for construction of a semigroup with generator $\Delta \phi$ in ${L^1}({R^N})$.References
- Philippe Benilan, Haim Brezis, and Michael G. Crandall, A semilinear equation in $L^{1}(R^{N})$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975), no. 4, 523–555. MR 390473
- Philippe Bénilan and Michael G. Crandall, The continuous dependence on $\varphi$ of solutions of $u_{t}-\Delta \varphi (u)=0$, Indiana Univ. Math. J. 30 (1981), no. 2, 161–177. MR 604277, DOI 10.1512/iumj.1981.30.30014
- Alan E. Berger, Haïm Brézis, and Joel C. W. Rogers, A numerical method for solving the problem $u_{t}-\Delta f(u)=0$, RAIRO Anal. Numér. 13 (1979), no. 4, 297–312 (English, with French summary). MR 555381, DOI 10.1051/m2an/1979130402971
- H. Brezis and A. Pazy, Semigroups of nonlinear contractions on convex sets, J. Functional Analysis 6 (1970), 237–281. MR 0448185, DOI 10.1016/0022-1236(70)90060-1
- H. Brézis and A. Pazy, Convergence and approximation of semigroups of nonlinear operators in Banach spaces, J. Functional Analysis 9 (1972), 63–74. MR 0293452, DOI 10.1016/0022-1236(72)90014-6
- Jean-Michel Coron, Formules de Trotter pour une équation d’évolution quasilinéaire du premier ordre, J. Math. Pures Appl. (9) 61 (1982), no. 1, 91–112 (French). MR 664343
- M. G. Crandall and T. M. Liggett, Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. Math. 93 (1971), 265–298. MR 287357, DOI 10.2307/2373376
- Tosio Kato and Kyûya Masuda, Trotter’s product formula for nonlinear semigroups generated by the subdifferentials of convex functionals, J. Math. Soc. Japan 30 (1978), no. 1, 169–178. MR 477931, DOI 10.2969/jmsj/03010169
- Shinnosuke Ôharu and Tadayasu Takahashi, A convergence theorem of nonlinear semigroups and its application to first order quasilinear equations, J. Math. Soc. Japan 26 (1974), 124–160. MR 341216, DOI 10.2969/jmsj/02610124
Bibliographic Information
- © Copyright 1984 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 92 (1984), 509-514
- MSC: Primary 35K55; Secondary 47H20
- DOI: https://doi.org/10.1090/S0002-9939-1984-0760935-8
- MathSciNet review: 760935