QUASI-LINEAR EVOLUTION AND ELLIPTIC EQUATIONS

Mikola Ivanovich Yaremenko

Mikola Ivanovich Yaremenko Department of Mathematics, National Technical University of Ukraine Kyiv Polytechnic Institute, Ukraine

Keywords: differential form, parabolic equations, evolution equations, a priori estimate, weak solution, singular coefficients

Abstract

In this article we use a new type of nonlinear elliptic operators that are associated with left side of elliptic equation and studied their properties. We draw up the form, that is associated with non-linear elliptic operator , studying the properties this operator by means of form.

We proved some a priori estimates which are theorems about properties of solutions under certain conditions on the function that forming this equation. We proved the existence of solution of quasi-linear evolution equation with singular coefficients in space by Galerkin method and showed that a given equationhas a solution in the Sobolev space.

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References

Brezis H., Pazy A. Semigroups of non-linear contractions on convex sets, J. Func. Anal,. 1970; V. 6, pp. 237-281.

Boccardo L., Giachetti D., Diaz J.-I., Murat F. Existence and regularity of renormalized solutions for some elliptic problems involving derivatives of nonlinear terms, Journal of Differential Equations, Vol. 106, N. 2, pp. 215-237, 1993.

Browder F.E. Existence of periodic solutions for nonlinear equations of evolution, Proc. Nat. Acad. Sci. USA, 1965; V. 53, pp. 1100-1103.

Blanchard D., Guibe O., Redwane H., Nonlinear equations with unbounded heat conduction and integrable data, Annali di Matematica Pura ed Applicata, Vol. 187, N. 3, pp. 405-433, 2008.

Bheeman Radhakrishnan, Aruchamy Mohanraj, Vinoba V. Existence of solutions for nonlinear impulsive neutral integro-differential equations of Sobolev type with nonlocal conditions in Banach spaces, Electronic Journal of Differential Equations, Vol. 2013 (2013), No. 18, pp. 1-13.

Redwane H. Existence of Solution for Nonlinear Elliptic Equations with Unbounded Coefficients and L1 Data, International Journal of Mathematics and Mathematical Sciences

Volume 2009 (2009), Article ID 219586, 18 P.

Kato T. Perturbation theory for linear operators, Berlin-Heidelberg-New York: Springer-Verlag, 1980; 578 p.

Komura Y. Differentiability of nonlinear semigroups, J. Math. Soc. Japan, 1969; V. 21, pp. 375-402.

Konishy Y. Cosine functions of operators in locally convex spaces, J. Fac. Sci. Univ. Tokyo, 1972; V. 18, pp. 443-463.

Ignat L., Zuazua E. Convergence rates for dispersive approximation schemes to nonlinear Schrodinger equations, J. Math. Pures Appl. (9) 98 (2012), N. 5, pp. 479-517,

Marica A., Zuazua E., On the quadratic finite element approximation of 1-d waves: propagation, observation and control, SIAM J. Numer. Anal., 50(5), 2012, pp. 2744-2777.

Yaremenko M.I. Field Equations in Space Y^n without symmetry conditions in the General Case, Global Journal of Physics Vol. 2, No. 1 July 08, 2015; pp.89-106.

Yaremenko M.I. The Stability of Solution of Cauchy Problem for Evolution System in Euclidean Space R^l, l>2 , Global Journal of Mathematics Vol. 7, No. 2 June 20, 2016; pp. 729-740.

Yaremenko M.I. Nonlinear elliptic operator in Sobolev space generated by elliptic equations, IOSR Journal of Mathematics. Volume 12, No. 3, 2016; pp. 62-73.

Yaremenko M.I. Global Holdler Continuity of Solutions of Quasilinear Differential Equations, Global Journal of Mathematics Vol. 7, No 1 June 14, 2016; pp. 711-728.

Tsutsumi Y. L2-solutions for nonlinear Schrodinger equations and nonlinear groups. Funkc. Ekvacioj, Ser. Int., 30:115-125, 1987.

Yajima K. Existence of solutions for Schrodinger evolution equations. Comm. Math. Phys., 110(3):415– 426, 1987.

Yin Y. F. On nonlinear parabolic equations with nonlocal boundary condition. J. Math. Anal. Appl. 185(1994), 161-174.