Hodge-type Decomposition for Time-dependent First-order Parabolic Operators with Non-constant Coefficients: The Variable Exponent Case

E. Acerbi and G. Mingione, Regularity results for stationary electro-rheological fluids, Arch. Ration. Mech. Anal., 164-No.3, 213–259 (2002)

R. Artino and J. Barros-Neto, Hypoelliptic Boundary-Value Problems, Lectures Notes in Pure and Applied Mathematics-Vol.53, Marcel Dekker, New York-Basel, 1980.

P. Cerejeiras, U. Kähler, M.M. Rodrigues and N. Vieira, Hodge-type decomposition in variable exponent spaces for the time-dependent operators: the Schrödinger case, Commun. Pure Appl. Anal., 13-No.6, (2014), 2253–2272

P. Cerejeiras and N. Vieira, Regularization of the non-stationary Schrödinger operator, Math. Meth. in Appl. Sc., 32-No.(4), (2009), 535–555

Cerejeiras P., Vieira N.: Factorization of the Non-Stationary Schrödinger Operator, Adv. Appl. Clifford Algebr. 17, 331–341 (2007)

P. Cerejeiras, U. Kähler and F. Sommen, Parabolic Dirac operators and the Navier- Stokes equations over time-varying domains, Math.Meth. in Appl. Sc.,28-No.14, (2005), 1715-1724.

Y. Chen, S. Levine and R. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66-No.4, (2006), 1383-1406.

R. Delanghe, F. Sommen and V. Souc̆ek, Clifford algebras and spinor-valued functions. A function theory for the Dirac operator, Mathematics and its Applications-Vol.53, Kluwer Academic Publishers, Dordrecht etc., 1992.

L. Diening, P. Harjulehto, P. Hästö and M. Ružička, Lebesgue and Sobolec Spaces with variable Exponents, Springer-Verlang, Berlin, 2011.

L. Diening, D. Lengeler and M. Ružička, The Stokes and Poisson problem in variable exponents spaces, Complex Var. Elliptic Equ., 56-No.7-9, (2011), 789-811.

R. Fortini, D. Mugnai and P. Pucci, Maximum principles for anisotropic elliptic inequalities, Nonlinear Anal.,70-No.8, (2009), 2917-2929.

K. Gürlebeck and W. Sprößig, Quaternionic and Clifford Calculus for Physicists and Engineers, Mathematical Methods in Practice, Wiley, Chichester, 1997.

R. F. Hoskins and J.S. Pinto, Theories of Generalised Functions - Distributions, Ultradistributions and other Generalised Functions, Horwood Publishing, Chichester, 2005.

O. Kováčik and J. Rákosník, On spaces L p(x) and W 1,p(x), Czechoslovak Math. J., 41(116)-No.4, (1991), 592-618.

R.S. Kraußhar, M.M. Rodrigues and N. Vieira, Hodge decomposition and solution formulas for some first-order time-dependent parabolic operators with non-constant coefficients,Ann. Mat. Pura Appl., in press, DOI: 10.1007/s10231-013-0357-3 .

R.S. Kraußhar and N. Vieira, The Schrödinger equation on cylinders and the n-torus, J. Evol. Equ., 11-No.1, (2011), 215-237.

Laskin N., Fractional Schrödinger equation. Phys. Rev. E, 66, 056108 (2002)

N. Laskin, Fractional quantum mechanics, Phy. Rev. E, 62-No.3, (2000), 3135-3145.

N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268, 298–305 (2000)

H. Nakano, Modulared Semi-Ordered Linear Spaces, Maruzen Co. Ltd., Tokyo, 1950.

H. Nakano, Topology of linear topological spaces, Maruzen Co. Ltd., Tokyo, 1951.

M. Sanchón and J.M. Urbano, Entropy solutions for the p(x)-Laplace equation Trans. Amer. Math. Soc., 361, (2009), 6387-6405.

S. Samko, On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators, Integr. Transf. Spec. F., 16-No.5-6, (2005), 461-482.

Sharapudinov I.I., Topology of the space L p(t)([0, t]). Math. Notes 26, 796-806 (1980)

W. Sprößig, On Helmotz decompositions and their generalizations-an overview, Math. Meth. in Appl. Sc., 33-No.4, (2009), 374-383.

T. Tao, Nonlinear dispersive equations. Local and global analysis, CBMS Regional Conference Series in Mathematics-Vol.106, American Mathematical Society, 2006.

G. Velo, Mathematical Aspects of the nonlinear Schrödinger Equation, Proceedings of the Euroconference on nonlinear Klein-Gordon and Schrdinger systems: theory and applications, L. Vázquez et al. (eds.), World Scientific, Singapore, (1996), 39-67.