# American Institute of Mathematical Sciences

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. [2] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Universitext, New York, 2010. [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. [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. [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. [6] E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. [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. [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. [9] G. Floridia, Controllability for nonlinear degenerate parabolic problems with Robin boundary conditions,, in preparation., (). [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. [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.

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. [2] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Universitext, New York, 2010. [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. [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. [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. [6] E. DiBenedetto, Degenerate Parabolic Equations, Springer-Verlag, New York, 1993. [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. [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. [9] G. Floridia, Controllability for nonlinear degenerate parabolic problems with Robin boundary conditions,, in preparation., (). [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. [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.
 [1] Doyoon Kim, Hongjie Dong, Hong Zhang. Neumann problem for non-divergence elliptic and parabolic equations with BMO$_x$ coefficients in weighted Sobolev spaces. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 4895-4914. doi: 10.3934/dcds.2016011 [2] Doyoon Kim, Kyeong-Hun Kim, Kijung Lee. Parabolic Systems with measurable coefficients in weighted Sobolev spaces. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022062 [3] Ping Li, Pablo Raúl Stinga, José L. Torrea. On weighted mixed-norm Sobolev estimates for some basic parabolic equations. Communications on Pure and Applied Analysis, 2017, 16 (3) : 855-882. doi: 10.3934/cpaa.2017041 [4] Li Ma. Blow-up for semilinear parabolic equations with critical Sobolev exponent. Communications on Pure and Applied Analysis, 2013, 12 (2) : 1103-1110. doi: 10.3934/cpaa.2013.12.1103 [5] Xiaojun Li, Xiliang Li, Kening Lu. Random attractors for stochastic parabolic equations with additive noise in weighted spaces. Communications on Pure and Applied Analysis, 2018, 17 (3) : 729-749. doi: 10.3934/cpaa.2018038 [6] Alberto Fiorenza, Anna Mercaldo, Jean Michel Rakotoson. Regularity and uniqueness results in grand Sobolev spaces for parabolic equations with measure data. Discrete and Continuous Dynamical Systems, 2002, 8 (4) : 893-906. doi: 10.3934/dcds.2002.8.893 [7] Chérif Amrouche, Mohamed Meslameni, Šárka Nečasová. Linearized Navier-Stokes equations in $\mathbb{R}^3$: An approach in weighted Sobolev spaces. Discrete and Continuous Dynamical Systems - S, 2014, 7 (5) : 901-916. doi: 10.3934/dcdss.2014.7.901 [8] Jun Yang, Yaotian Shen. Weighted Sobolev-Hardy spaces and sign-changing solutions of degenerate elliptic equation. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2565-2575. doi: 10.3934/cpaa.2013.12.2565 [9] Tahar Z. Boulmezaoud, Amel Kourta. Some identities on weighted Sobolev spaces. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 427-434. doi: 10.3934/dcdss.2012.5.427 [10] Claudia Anedda, Giovanni Porru. Boundary estimates for solutions of weighted semilinear elliptic equations. Discrete and Continuous Dynamical Systems, 2012, 32 (11) : 3801-3817. doi: 10.3934/dcds.2012.32.3801 [11] Daniel Coutand, Steve Shkoller. Turbulent channel flow in weighted Sobolev spaces using the anisotropic Lagrangian averaged Navier-Stokes (LANS-$\alpha$) equations. Communications on Pure and Applied Analysis, 2004, 3 (1) : 1-23. doi: 10.3934/cpaa.2004.3.1 [12] Hua Chen, Huiyang Xu. Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 1185-1203. doi: 10.3934/dcds.2019051 [13] Peter Weidemaier. Maximal regularity for parabolic equations with inhomogeneous boundary conditions in Sobolev spaces with mixed $L_p$-norm. Electronic Research Announcements, 2002, 8: 47-51. [14] Younghun Hong, Yannick Sire. On Fractional Schrödinger Equations in sobolev spaces. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2265-2282. doi: 10.3934/cpaa.2015.14.2265 [15] Flank D. M. Bezerra, Jacson Simsen, Mariza Stefanello Simsen. Convergence of quasilinear parabolic equations to semilinear equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3823-3834. doi: 10.3934/dcdsb.2020258 [16] Jiabao Su, Rushun Tian. Weighted Sobolev embeddings and radial solutions of inhomogeneous quasilinear elliptic equations. Communications on Pure and Applied Analysis, 2010, 9 (4) : 885-904. doi: 10.3934/cpaa.2010.9.885 [17] Wenxiong Chen, Chao Jin, Congming Li, Jisun Lim. Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations. Conference Publications, 2005, 2005 (Special) : 164-172. doi: 10.3934/proc.2005.2005.164 [18] T. V. Anoop, Nirjan Biswas, Ujjal Das. Admissible function spaces for weighted Sobolev inequalities. Communications on Pure and Applied Analysis, 2021, 20 (9) : 3259-3297. doi: 10.3934/cpaa.2021105 [19] Takesi Fukao, Masahiro Kubo. Nonlinear degenerate parabolic equations for a thermohydraulic model. Conference Publications, 2007, 2007 (Special) : 399-408. doi: 10.3934/proc.2007.2007.399 [20] Young-Sam Kwon. Strong traces for degenerate parabolic-hyperbolic equations. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1275-1286. doi: 10.3934/dcds.2009.25.1275

Impact Factor: