advertisement

Kingdom of Saudi Arabia Majmaah University Deanship of Scientific Research Engineering and Applied Sciences Research Center Research Project Grant Project No. 15 (2014-2015) Research Title: Evolution Equations on the Heinsenberg Group Duration of the project grant : 10 months Total Accorded Budget: 22000 SR Principal Researcher: Dr Rabah Kellil Associated Researcher: Pr Dr Mokhtar Kirane Department: : Mathematics College: Of Science Al-Zulfi Mobile: +966564070512 Official E-mail: [email protected] Dr Rabah Kellil * Abstract The Heisenberg group plays an important role in many branches of physics, mathematics, such as number theory, representation theory, several complex variables, quantum mechanics and Partial Differential Equations of any type (elliptic, parabolic, hyperbolic, ultra-parabolic or ultra-hyperbolic). We will study some evolution equations with fractional derivatives in time, we intend to present some results concerning nonexistence of global solution which indicate that some solutions become infinite in a finite time revealing some singularities. Our equations may contain one or may derivatives in time; they may interpolate between the heat equation and the wave equation. Our equations may contain many times as in physics some models contain two or may times. Moreover, we will extend the study on one equation to a system of two equations strongly coupled via the non-linear coupling terms. Our method of proof will relay on the weak formulation of the sought solution with a judicious choice of the test function and a scaling argument. Pr Dr Mokhtar Kirane * * Expected Results Our objectives in studying non-global existence (blow-up) of solutions to evolution equations is to capture their essential futures; a blowing-up solutions may tell two things: either there is a physical singularity or the non-linearity taken in the model is not suitable and hence the model has to be changed. Our objectives are clear: 1-Obtain the conditions leading to non-existence solutions (which in presence of a local existence theorem means blow-up) of some evolutions equations. 2- Extend the results to systems of at least two equations. 3- Extend the study to equations with at least two times. 4- Extend the study to equations with fractional time derivatives. 5- Extend the study to systems with fractional time derivatives. Conclusions Introduction In particles physics, it was proved by Weyl that the Schrodinger picture and the Heisenberg picture are physically equivalent. The Heisenberg picture is based on the Heisenberg group. From here emerged a large number of problems in many fields: physics, mathematics, number theory, representation theory, several complex variables, quantum mechanics and Partial Differential Equations. In partial differential equation, all articles deal with equations with usual derivatives. In recent years, appears the necessity to study evolutions equations (and even stationary equations) with derivatives of non-integer orders. Most of the articles deal with linear equations whose studies are easy as they are performed using the Laplace transform. Our aim is to tackle non-linear equations with many (at least two) fractional derivatives on one hand and non-linear evolutions equations with multi-times on another hand. The main problem with non-linear equations is that solutions may experience blow-up in a finite time. The eventual non-boundedness of the sought solutions depends on the non- linearity, the dimension of the space and the underlying functional space in which the equation is studied. In our project, we will study some evolution equations with or without fractional derivatives in time; we intend to present some results concerning nonexistence of global solution which indicate that some solutions become infinite in a finite time revealing some singularities in the physical problem. Our equations may contain one or many derivatives in time; they may interpolate between the heat equation and the wave equation. Also our equations may contain many times as in physics some models contain two or many times. Moreover, we will extend the study on one equation to a system of two equations strongly coupled via the non-linear coupling terms. Our method of proof will relay on the weak formulation of the sought solution with a judicious choice of the test function and a scaling argument. Our results will shed light not only on the physics of the problems but also on the methods of study. Literature Review Non-autonomous evolutions equations with at least two fractional derivatives in time and even a fractional derivative in but linear have been studied in a series of papers by R. K. Saxena, A. M. Mathai and H. J. Haubold using the Laplace transform. They obtained explicit representation of solutions. These type of equations appear naturally in engineering as they describe hereditary phenomena or memory effects. They also appear in several other complex systems such as atmospheric diffusion of pollution, traffic network, anomalous diffusion ( in rocks for example: oil modeling), biology, electronics, wave propagation (as in geophysics for example), mechanics, astrophysics, signal and image processing, optics, financial engineering, etc. Multi-time models appear in theoretical physics and Multi-time models with fractional derivatives appear in physics as time is fractal in essence. Methods and Materials We will use modern functional analysis: fixed point theorems when dealing with local existence theorems, a priori estimates when we look for global solutions, and the weak formulation of the problems with a judicious choice of the test function and a scaling argument when we want to show non-existence results. Planned processes of major tasks as follow: 1- Non-existence results for one equation with two fractional derivatives 2 Non-existence results for one system with two equations and two fractional derivatives 3-Non-existence results for one equation posed on the Heisenberg group with two fractional derivatives 4-Non-existence results for one system with two equations posed on the Heisenberg group with two fractional derivatives References 1- R. K. Saxena, A. M. Mathai and H. J. Haubold, Reaction-Diffusion Systems and Nonlinear waves, Astrophysics Space Sciences (2006) 305:297-303. 2-R. K. Saxena, A. M. Mathai and H. J. Haubold, On generalized fractional kinetic equations, Physica A (2006) 344: 657-664. 3-R. L. Bagley and P. J. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheology (1983) 27: 201-210. 4-R. L. Bagley and P. J. Torvik, On the appearance of the fractional derivative in the behavior of real materials, J. Appl. Mechanics (1983) 51: 294-298. 4-R. Hilfer, Fractional time evolution, applications of fractional calculus in Physics, World Scientific, N.J., London, H.K, 2000. 5-A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies 204, 2006. 6-I. Podlubny, Fractional Differential Equations, Mathematics in Sciences and Engineering, 198, Academic Press, 1999. 7-M. Kirane, Qualitative properties of solutions to a time-space fractional evolution equation. Quarterly in Applied Mathematics (2012) 70: 133-157. 8-M. Kirane, The profile of blowing-up solutions to a nonlinear system of fractional differential equations. Nonlinear Analysis, T. M. A, (2010): 3723-3736. 9-M. Kirane, Semilinear Volterra integro-differential problems with fractional derivatives in the nonlinearities. Abstract and Applied Mathematics 2011, Art. ID 510314, 11p. 10-M. Kirane, Critical exponents of Fujita type for certain evolution equations and systems with spatio-temporal fractional derivatives, J. Math. Anal. Appl. (2005) 312: 488-501. 11-M. Kirane, Image structure preserving denoising using generalized fractional time integrals. Signal Processing (2012) 92(2): 553-563. 12-M. Kirane, Determination of an unknown source term and the temperature distribution for the linear heat equation involving fractional derivative in time. Applied Mathematics and Computation (2011) 218(1): 163-170. 13-M. Kirane, An inverse source problem for a two dimensional time fractional diffusion equation with non-local boundary condition. Math. Methods in the Applied Sciences, DOI; 10. 1002/MMAS: 2161. 14-Bars I., Survey of two-time physics, Class. Quant. Grav., 18 (2001), pp. 3113-3130. 15-Barashenkov V. S., Mechanics in Six-Dimensional Space-time, Foundations of Physics, Vol. 28, No 3, 1988, pp. 471-484. 16- Barashenkov V. S., Propagation of signals in space with multi-dimensional time, JINR, E2-96-112, Dubna, 1996. 17- Brenner A., The mixed problems in several time variables and problems with parameters. International Conference on Differential Equations, River Edge, NJ, Vol. 80, no. 9, pp. 901-918, 2001. 18-Courant R. and D. Hilbert, Methods of Mathematical Physics, Vol II: Partial Differential Equations. Inter-science, New York, 1962. 19- Craig, W. and S. Weinstein, On determinism and well-posedness in multiple time dimensions. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 465 (2009), no. 2110, 3023–3046. 20-Dorling J., The dimensionality of time, Am. J. Phys., 38 (1970), pp. 539-540. * Dr Rabah Kellil , Department of Mathematics, College of Science Al-Zulfi Majmaah University, KSA. ** Pr Dr Mokhtar Kirane, Laboratoire de Mathématiques, Image et Applications, Pôle Scientifique et Technologique, Université de La Rochelle, Av:.M. Crépeau, 17000 La Rochelle, France.