Semilinear evolution equations in Banach spaces with application to parabolic partial differential equations

Rankin, Samuel M.

doi:10.1090/S0002-9947-1993-1052911-5

Semilinear evolution equations in Banach spaces with application to parabolic partial differential equations
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by Samuel M. Rankin

Trans. Amer. Math. Soc. 336 (1993), 523-535

DOI: https://doi.org/10.1090/S0002-9947-1993-1052911-5

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Abstract:

A theory for a class of semilinear evolution equations in Banach spaces is developed which when applied to certain parabolic partial differential equations with nonlinear terms in divergence form gives strong solutions even for nondifferentiable data.

References

Lamberto Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes, Rend. Sem. Mat. Univ. Padova 31 (1961), 308–340 (Italian). MR 138894
Thomas J. Bridges, The Hopf bifurcation with symmetry for the Navier-Stokes equations in $(L_p(\Omega ))^n,$ with application to plane Poiseuille flow, Arch. Rational Mech. Anal. 106 (1989), no. 4, 335–376. MR 997106, DOI 10.1007/BF00281352
Avner Friedman, Partial differential equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. MR 0445088
Daisuke Fujiwara and Hiroko Morimoto, An $L_{r}$-theorem of the Helmholtz decomposition of vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 (1977), no. 3, 685–700. MR 492980
Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR 610244, DOI 10.1007/BFb0089647
Melvin L. Heard and Samuel M. Rankin III, Weak solutions for a class of parabolic Volterra integrodifferential equations, J. Math. Anal. Appl. 139 (1989), no. 1, 78–109. MR 991928, DOI 10.1016/0022-247X(89)90231-X
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486, DOI 10.1007/978-1-4612-5561-1
Roger Temam, Infinite-dimensional dynamical systems in mechanics and physics, Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1988. MR 953967, DOI 10.1007/978-1-4684-0313-8
H. Triebel, Interpolation theory, function spaces, differential operators, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978. MR 500580
Fred B. Weissler, Semilinear evolution equations in Banach spaces, J. Functional Analysis 32 (1979), no. 3, 277–296. MR 538855, DOI 10.1016/0022-1236(79)90040-5
Kôsaku Yosida, Functional analysis, 6th ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 123, Springer-Verlag, Berlin-New York, 1980. MR 617913