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, 1 (1992).   Google Scholar

[2]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Springer-Universitext, (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, (2011), 19.   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 (2011), 1.  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 (2010), 1293.   Google Scholar

[6]

E. DiBenedetto, Degenerate Parabolic Equations,, Springer-Verlag, (1993).   Google Scholar

[7]

G. Floridia, Approximate controllability for nonlinear degenerate parabolic problems with bilinear control,, J. Differential Equations, 257 (2014), 3382.  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, (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.  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 (2012), 63.   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, 1 (1992).   Google Scholar

[2]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Springer-Universitext, (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, (2011), 19.   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 (2011), 1.  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 (2010), 1293.   Google Scholar

[6]

E. DiBenedetto, Degenerate Parabolic Equations,, Springer-Verlag, (1993).   Google Scholar

[7]

G. Floridia, Approximate controllability for nonlinear degenerate parabolic problems with bilinear control,, J. Differential Equations, 257 (2014), 3382.  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, (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.  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 (2012), 63.   Google Scholar

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