In this paper, the existence and uniqueness of local solutions to the initial and boundary value problem of a class of parabolic system related to the p-Laplacian are studied. The regularization method is used to construct a sequence of approximation solutions, with the help of monotone iteration technique, then we get the existence of solution of a regularized system. By the use of a standard limiting process, the existence of the local solutions of the system is obtained. Finally, the uniqueness of the solution is also proven.
References
[1]
Astrita, G. and Marrucci, G. (1974) Principles of Non-Newtonian Fluid Mechanics. McGraw-Hill, New York.
[2]
Martinson, L.K. and Pavlov, K.B. (1971) Unsteady Shear Flows of a Conducting Fluid with a Rheological Power Law. Magnitnaya Gidrodinamika, 2, 50-58.
[3]
Esteban, J.R. and Vazquez, J.L. (1982) On the Equation of Turbulent Filteration in One-Dimensional Porous Media. Nonlinear Analysis, 10, 1303-1325. http://dx.doi.org/10.1016/0362-546X(86)90068-4
[4]
Constantin, A., Escher, J. and Yin, Z. (2004) Global Solutions for Quasilinear Parabolic System. Journal of Differential Equations, 197, 73-84. http://dx.doi.org/10.1016/S0022-0396(03)00165-7
[5]
Dichstein, F. and Escobedo, M. (2001) A Maximum Principle for Semilinear Parabolic Systems and Application. Nonlinear Analysis, 45, 825-837. http://dx.doi.org/10.1016/S0362-546X(99)00419-8
[6]
Pierre, M. and Schmidt, D. (1997) Blowup in Reaction-Diffusion Systems with Dissipation of Mass. SIAM Journal on Mathematical Analysis, 28, 259-269. http://dx.doi.org/10.1137/S0036141095295437
[7]
Zhao, J. (1993) Existence and Nonexistence of Solutions for . Journal of Mathematical Analysis and Applications, 172, 130-146. http://dx.doi.org/10.1006/jmaa.1993.1012
[8]
Wei, Y. and Gao, W. (2007) Existence and Uniqueness of Local Solutions to a Class of Quasilinear Degenerate Parabolic Systems. Applied Mathematics and Computation, 190, 1250-1257. http://dx.doi.org/10.1016/j.amc.2007.02.007
[9]
Ladyzenskaja, O.A., Solonnikov, V.A. and Ural’ceva, N.N. (1968) Linear and Quasilinear Equations of Parabolic Type. American Mathematical Society, Providence, RI.
[10]
Friedman, A. (1964) Partial Differential Equations of Parabilic Type. Prentice-Hall Inc., Englewood Cliffs, NJ.
[11]
Coddingtin, E. and Levinson, N. (1955) Theory of Ordinary Differential Equations. McGraw-Hill, New York.
