2015, 2015(special): 455-463. doi: 10.3934/proc.2015.0455

Well-posedness for a class of nonlinear degenerate parabolic equations

1. 

Post Doc Istituto Nazionale di Alta Matematica (INdAM) "F. Severi", Dipartimento di Matematica, Università di Roma "Tor Vergata", I-00161 Roma, Italy

Received  September 2014 Revised  August 2015 Published  November 2015

In this paper we obtain well-posedness for a class of semilinear weakly degenerate reaction-diffusion systems with Robin boundary conditions. This result is obtained through a Gagliardo-Nirenberg interpolation inequality and some embedding results for weighted Sobolev spaces.
Citation: Giuseppe Floridia. Well-posedness for a class of nonlinear degenerate parabolic equations. Conference Publications, 2015, 2015 (special) : 455-463. doi: 10.3934/proc.2015.0455
References:
[1]

A. Bensoussan, G. Da Prato, G. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, 1, Birkäuser, Boston, 1992. Google Scholar

[2]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Universitext, New York, 2010. Google Scholar

[3]

P. Cannarsa and G. Floridia, Approximate controllability for linear degenerate parabolic problems with bilinear control, Proc. Evolution Equations and Materials with Memory 2010 (eds. D. Andreucci, S. Carillo, M. Fabrizio, P. Loreti and D. Sforza), Universitá La Sapienza Roma, (2011), 19-36, arXiv:1106.4232.  Google Scholar

[4]

P. Cannarsa and G. Floridia, Approximate multiplicative controllability for degenerate parabolic problems with Robin boundary conditions, Communications in Applied and Industrial Mathematics, 2(2) (2011), 1-16. doi: 10.1685/journal.caim.376.  Google Scholar

[5]

P. Cannarsa and A. Y. Khapalov, Multiplicative controllability for reaction-diffusion equations with target states admitting finitely many changes of sign, Discrete Contin. Dyn. Syst. Ser. B, 14(4) (2010), 1293-1311. Google Scholar

[6]

E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. Google Scholar

[7]

G. Floridia, Approximate controllability for nonlinear degenerate parabolic problems with bilinear control, J. Differential Equations, 257(9) (2014), 3382-3422, arXiv:1406.1447. doi: 10.1016/j.jde.2014.06.016.  Google Scholar

[8]

G. Floridia, Approximate Multiplicative Controllability for Degenerate Parabolic Problems and Regularity Properties of Elliptic and Parabolic Systems, Ph.D thesis, University of Catania, Supervisor: Prof. Piermarco Cannarsa, 2011. Google Scholar

[9]

G. Floridia, Controllability for nonlinear degenerate parabolic problems with Robin boundary conditions,, in preparation., ().   Google Scholar

[10]

G. Floridia and M. A. Ragusa, Interpolation inequalities for weak solutions of nonlinear parabolic systems, J. Inequal. Appl., 42 (2011), 1-17. doi: 10.1186/1029-242X-2011-42.  Google Scholar

[11]

G. Floridia and M. A. Ragusa, Differentiabilty and partial Hölder continuity of solutions of nonlinear elliptic systems, J. Convex Anal., 19(1) (2012), 63-90. Google Scholar

show all references

References:
[1]

A. Bensoussan, G. Da Prato, G. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, 1, Birkäuser, Boston, 1992. Google Scholar

[2]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Universitext, New York, 2010. Google Scholar

[3]

P. Cannarsa and G. Floridia, Approximate controllability for linear degenerate parabolic problems with bilinear control, Proc. Evolution Equations and Materials with Memory 2010 (eds. D. Andreucci, S. Carillo, M. Fabrizio, P. Loreti and D. Sforza), Universitá La Sapienza Roma, (2011), 19-36, arXiv:1106.4232.  Google Scholar

[4]

P. Cannarsa and G. Floridia, Approximate multiplicative controllability for degenerate parabolic problems with Robin boundary conditions, Communications in Applied and Industrial Mathematics, 2(2) (2011), 1-16. doi: 10.1685/journal.caim.376.  Google Scholar

[5]

P. Cannarsa and A. Y. Khapalov, Multiplicative controllability for reaction-diffusion equations with target states admitting finitely many changes of sign, Discrete Contin. Dyn. Syst. Ser. B, 14(4) (2010), 1293-1311. Google Scholar

[6]

E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. Google Scholar

[7]

G. Floridia, Approximate controllability for nonlinear degenerate parabolic problems with bilinear control, J. Differential Equations, 257(9) (2014), 3382-3422, arXiv:1406.1447. doi: 10.1016/j.jde.2014.06.016.  Google Scholar

[8]

G. Floridia, Approximate Multiplicative Controllability for Degenerate Parabolic Problems and Regularity Properties of Elliptic and Parabolic Systems, Ph.D thesis, University of Catania, Supervisor: Prof. Piermarco Cannarsa, 2011. Google Scholar

[9]

G. Floridia, Controllability for nonlinear degenerate parabolic problems with Robin boundary conditions,, in preparation., ().   Google Scholar

[10]

G. Floridia and M. A. Ragusa, Interpolation inequalities for weak solutions of nonlinear parabolic systems, J. Inequal. Appl., 42 (2011), 1-17. doi: 10.1186/1029-242X-2011-42.  Google Scholar

[11]

G. Floridia and M. A. Ragusa, Differentiabilty and partial Hölder continuity of solutions of nonlinear elliptic systems, J. Convex Anal., 19(1) (2012), 63-90. Google Scholar

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