# Nonlinear Parabolic Operators with Perturbed Coefficients

## DOI:

https://doi.org/10.26713/cma.v9i3.790## Keywords:

Nonlinear parabolic equations, Cauchy-Dirichlet problem, VMO, Implicit Function Theorem, Newton Iteration Procedure## Abstract

We consider the Cauchy-Dirichlet problem for second order quasilinear non-divergence form parabolic equations with discontinuous data in a bounded cylinder \(Q\). Supposing existence of strong solution \(u_0\) and applying the Implicit Function Theorem we show that for any small \(L^\infty\)-perturbation of the coefficients there exists, locally in time, exactly one solution \(u\) close to $u_0$ with respect to the norm in \(W^{2,1}_p(Q)\) which depends smoothly on the data. For that, no structure and growth conditions on the data are needed. Moreover, applying the Newton Iteration Procedure we obtain an approximating sequence for the solution \(u_0\).### Downloads

## References

P. Acquistapace, On BMO regularity for linear elliptic systems, Ann. Mat. Pura. Appl. 161 (1992), 231 – 270, DOI: 10.1007BF01759640.

M. Bramanti and M.C. Cerutti, W1,2 p solvability for the Cauchy–Dirichlet problem for parabolic equations with VMO coefficients, Comm. Partial Diff. Eq. 18 (1993), 1735 – 1763, DOI: 10.1080/0360530930882991.

D. Cassani, L. Fattorusso and A. Tarsia, Global existence for nonlocal MEMS, Nonl. Anal., Theory Meth. Appl., Ser. A. 74 (16) (2011), 5722 – 5726, DOI: 10.1007/j.na.2011.05.060.

J.A. Griepentrog and L. Recke, Local existence, uniqueness and smooth dependence for nonsmooth quasilinear parabolic problems, J. Evol. Eq. 10 (2010), 341 – 375, DOI: 10.1007/s00028-010-0052-4.

K. Gröger and L. Recke, Applications of differential calculus tu quasilinear elliptic boundary value problems with non-smooth data, Nonl. Differ. Equ. Appl. 13 (3) (2006), 263 – 285, DOI: 10.1007/s00030-006-3017-0.

F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415 – 426, DOI: 10.1002/cpa.3160140317.

P.W. Jones, Extension theorems for BMO, Indiana Univ. Math. J. 29 (1980), 41 – 66.

O.A. Ladyzhenskaya, V.A. Solonnikov and N.N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monographs 23, Amer. Math. Soc., Providence, R.I. (1968).

O.A. Ladyzhenskaya and N.N. Ural'tseva, A survey of results on the solubility of boundary-value problems for second-order uniformly elliptic and parabolic quasi-linear equations having unbounded singularities, Russ. Math. Surv. 41 (5) (1986), 1 – 31, DOI: 10.1070/RM1986v041n05ABEH003415.

A. Maugeri, D.K. Palagachev and L.G. Softova, Elliptic and Parabolic Equations with Discontinuous Coefficients, Math. Res. 109, Wiley-VCH, Berlin (2000), DOI: 10.1002/3527600868.

D.K. Palagachev, L. Recke and L.G. Softova, Applications of the differential calculus to nonlinear elliptic operators with discontinuous coefficients, Math. Ann. 336 (2006), 617 – 637, DOI: 10.1007/s00208-006-0014-X.

L. Recke, Applications of the implicit function theorem to quasilinear elliptic boundary value problem with non-smooth data, Comm. Part. Diff. Eq. 20 (1995), 1457 – 1479, DOI: 10.1080/03605309508821140.

L. Recke and L.G. Softova, Application of the differential calculus to nonlinear parabolic operators, C. R. Acad. Bulg. Sci. 66 (2) (2013), 185 – 192, DOI: 10.7546/CR-2013-66-2-13101331-4.

D. Sarason, Functions of vanishing mean oscillation, Trans. Amer. Math. Soc. 207 (1975), 391 – 405, DOI: 10.2307/1997184.

L.G. Softova, Quasilinear parabolic operators with discontinuous ingredients, Nonl. Anal., Theory Meth. Appl., Ser. A. 52 (2003), 1079 – 1093, DOI: 10.1016/s0362-546X(02)00128-1.

L.G. Softova, Strong solvability for a class of nonlinear parabolic equations, Le Matematiche, LII (1) (1997), 59 – 70, ISSN: 0373-3505.

L.G. Softova, An integral estimate for the gradient for a class of nonlinear elliptic equations in the plane, Z. Anal. Anwend. 17 (1) (1998), 57 – 66, DOI: 10.4171/ZAA/808.

A. Tarsia, Differential equations and implicit function: a generalization of the near operators theorem, Topol. Meth. Nonl. Anal. 11 (1998), 115 – 133, DOI: 10.12775/TMNA.1998.007.

E. Zeidler, Nonlinear Functional Analysis and its Applications, Vol. I: Fixed Points Theorems, Springer, Berlin ” Heidelberg ” New York (1993).

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*Communications in Mathematics and Applications*,

*9*(3), 277–292. https://doi.org/10.26713/cma.v9i3.790

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