Solvability and maximal regularity results for a differential equation with diffusion coefficient

We consider a second-order differential equation with rapidly growing intermediate coefficients. We obtain a solvability result in the cases that the diffusion coefficient of equation is unbounded or it tends to zero at the infinity. Under additional conditions, we prove the $L_p - $ maximal regularity estimate for the solution of this equation.

___

  • [1] Amann H. Quasilinear parabolic functional evolution equations. In: Chipot M, Ninomiya H (editors). Recent Avdances in Elliptic and Parabolic Issues. River Edge, NJ, USA: World Scientific, 2006, pp. 19-44.
  • [2] Bogachev VI, Krylov NV, Röckner M, Shaposhnikov SV. Fokker-Planck-Kolmogorov equations. American Mathematical Society. Mathematical Surveys and Monographs 2015; 207.
  • [3] Coddington EA, Levinson N. Theory of Ordinary Differential Equations. New York, NY, USA: McGraw-Hill, 1955.
  • [4] Da Prato G, Zabczyk J. Stochastic Equations in Infinite Dimensions. Cambridge, UK: Cambridge University Press, 1992.
  • [5] Denk R, Hieber M, Prüss J. R-boundedness, Fourier multipliers and problems of elliptic and parabolic type. Memoirs of the American Mathematical Society 2003; 166 (788): 1-114.
  • [6] Fornaro S, Lorenzi L. Generation results for elliptic operators with unbounded diffusion coefficients in L p -and Cb -spaces. Discrete & Continuous Dynamical Systems 2007; 18 (4): 747-772.
  • [7] Hieber M, Lorenzi L, Prüss J, Rhandi A, Schnaubelt R. Global properties of generalized Ornstein–Uhlenbeck operators on Lp(R N , R N ) with more than linearly growing coefficients. Journal of Mathematical Analysis and Applications 2009; 350 (1): 100-121.
  • [8] Hieber S., Sawada O. The Navier-Stokes Equations in Rn with Linearly Growing Initial Data. Archive for Rational Mechanics and Analysis 2005; 175: 269-285.
  • [9] Kato T. Perturbation Theory for Linear Operators. Berlin, Germany: Springer-Verlag, 1995.
  • [10] Metafune G, Pallara D, Vespri V. L p -estimates for a class of elliptic operators with unbounded coefficients in R n . Houston Journal of Mathematics 2005; 31: 605-620.
  • [11] Muckenhoupt B. Hardy’s inequality with weights. Studia Mathematica 1972; 44 (1): 31-38.
  • [12] Naimark MA. Linear differential operators. New York, NY, USA: Dover Publications, 2014.
  • [13] Ospanov KN. L1 -maximal regularity for quasilinear second order differential equation with damped term. Electronic Journal of Qualitative Theory of Differential Equations 2015; 39: 1-9.
  • [14] Ospanov KN. Maximal Lp -regularity for a second-order differential equation with unbounded intermediate coefficient. Electronic Journal of Qualitative Theory of Differential Equations 2019; 65: 1-13.
  • [15] Richtmyer RD. Principles of advanced mathematical physics. Vol.1. New York, NY, USA: Springer, 1978.
  • [16] Yosida K. Functional analysis. Berlin, Germany: Springer-Verlag, 1995.