# American Institute of Mathematical Sciences

May  2017, 22(3): 877-893. doi: 10.3934/dcdsb.2017044

## Global existence and regularity results for strongly coupled nonregular parabolic systems via iterative methods

 Department of Mathematics, University of Texas at San Antonio, One UTSA Circle, San Antonio, TX 78249, USA

* Corresponding author

Dedicated to Professor Stephen Cantrell on the occasion of his 60th birthday

Received  September 2015 Revised  December 2015 Published  January 2017

The global existence of classical solutions to strongly coupled parabolic systems is shown to be equivalent to the availability of an iterative scheme producing a sequence of solutions with uniform continuity in the BMO norms. Amann's results on global existence of classical solutions still hold under much weaker condition that their BMO norms do not blow up in finite time. The proof makes use of some new global and local weighted Gagliardo-Nirenberg inequalities involving BMO norms.

Citation: Dung Le. Global existence and regularity results for strongly coupled nonregular parabolic systems via iterative methods. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 877-893. doi: 10.3934/dcdsb.2017044
##### References:
 [1] S. Ahmad and D. Le, Global and blow up solutions to cross diffusion systems, Adv. Nonlinear Anal., 4 (2015), 209-219.  doi: 10.1515/anona-2015-0023.  Google Scholar [2] H. Amann, ynamic theory of quasilinear parabolic equations Ⅱ. Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75.   Google Scholar [3] H. Amann, Dynamic theory of quasilinear parabolic systems Ⅲ. Global existence, Math Z., 202 (1989), 219-250.  doi: 10.1007/BF01215256.  Google Scholar [4] B. Franchi, C. Perez and R. L. Wheeden, Self-improving properties of John Nirenberg and Poincaré inequalities on spaces of homogeneous type, J. Functional Analysis, 153 (1998), 108-146.  doi: 10.1006/jfan.1997.3175.  Google Scholar [5] A. Friedman, Partial Differential Equations, New York, 1969. Google Scholar [6] M. Giaquinta and M. Struwe, On the partial regularity of weak solutions of nonlinear parabolic systems, Math. Z., 179 (1982), 437-451.  doi: 10.1007/BF01215058.  Google Scholar [7] E. Giusti, Direct Methods in the Calculus of Variations World Scientific, 2003. Google Scholar [8] R. L. Johnson and C. J. Neugebauer, Properties of BMO functions whose reciprocals are also BMO, Z. Anal. Anwendungen, 12 (1993), 3-11.  doi: 10.4171/ZAA/583.  Google Scholar [9] D. Le, Regularity of BMO weak solutions to nonlinear parabolic systems via homotopy, Trans. Amer. Math. Soc., 365 (2013), 2723-2753.  doi: 10.1090/S0002-9947-2012-05720-5.  Google Scholar [10] D. Le, Global existence results for near triangular nonlinear parabolic systems, Adv. Nonlinear Studies, 13 (2013), 933-944.  doi: 10.1515/ans-2013-0410.  Google Scholar [11] D. Le, L. Nguyen and T. Nguyen, Coexistence in Cross Diffusion systems, Indiana Univ. J. Math., 56 (2007), 1749-1791.   Google Scholar [12] D. Le, Weighted Gagliardo-Nirenberg inequalities involving BMO norms and solvability of strongly coupled prabolic systems, Adv. Nonlinear Studies, 16 (2016), 125-146.  doi: 10.1515/ans-2015-5006.  Google Scholar [13] J. Orobitg and C. Pérez, $A_p$ weights for nondoubling measures in ${\rm{R}}^n$ and applications, Trans. Amer. Math. Soc., 354 (2002), 2013-2033.  doi: 10.1090/S0002-9947-02-02922-7.  Google Scholar [14] N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theor. Biol., 79 (1979), 83-99.  doi: 10.1016/0022-5193(79)90258-3.  Google Scholar [15] E. M. Stein, Harmonic Analysis, Real Variable Methods, Orthogonality and Oscillatory Integrals, Princeton Univ. Press, Princeton, NJ, 1993. Google Scholar [16] P. Strzelecki, Gagliardo Nirenberg inequalities with a BMO term, Bull. London Math. Soc., 38 (2006), 294-300.  doi: 10.1112/S0024609306018169.  Google Scholar

show all references

##### References:
 [1] S. Ahmad and D. Le, Global and blow up solutions to cross diffusion systems, Adv. Nonlinear Anal., 4 (2015), 209-219.  doi: 10.1515/anona-2015-0023.  Google Scholar [2] H. Amann, ynamic theory of quasilinear parabolic equations Ⅱ. Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75.   Google Scholar [3] H. Amann, Dynamic theory of quasilinear parabolic systems Ⅲ. Global existence, Math Z., 202 (1989), 219-250.  doi: 10.1007/BF01215256.  Google Scholar [4] B. Franchi, C. Perez and R. L. Wheeden, Self-improving properties of John Nirenberg and Poincaré inequalities on spaces of homogeneous type, J. Functional Analysis, 153 (1998), 108-146.  doi: 10.1006/jfan.1997.3175.  Google Scholar [5] A. Friedman, Partial Differential Equations, New York, 1969. Google Scholar [6] M. Giaquinta and M. Struwe, On the partial regularity of weak solutions of nonlinear parabolic systems, Math. Z., 179 (1982), 437-451.  doi: 10.1007/BF01215058.  Google Scholar [7] E. Giusti, Direct Methods in the Calculus of Variations World Scientific, 2003. Google Scholar [8] R. L. Johnson and C. J. Neugebauer, Properties of BMO functions whose reciprocals are also BMO, Z. Anal. Anwendungen, 12 (1993), 3-11.  doi: 10.4171/ZAA/583.  Google Scholar [9] D. Le, Regularity of BMO weak solutions to nonlinear parabolic systems via homotopy, Trans. Amer. Math. Soc., 365 (2013), 2723-2753.  doi: 10.1090/S0002-9947-2012-05720-5.  Google Scholar [10] D. Le, Global existence results for near triangular nonlinear parabolic systems, Adv. Nonlinear Studies, 13 (2013), 933-944.  doi: 10.1515/ans-2013-0410.  Google Scholar [11] D. Le, L. Nguyen and T. Nguyen, Coexistence in Cross Diffusion systems, Indiana Univ. J. Math., 56 (2007), 1749-1791.   Google Scholar [12] D. Le, Weighted Gagliardo-Nirenberg inequalities involving BMO norms and solvability of strongly coupled prabolic systems, Adv. Nonlinear Studies, 16 (2016), 125-146.  doi: 10.1515/ans-2015-5006.  Google Scholar [13] J. Orobitg and C. Pérez, $A_p$ weights for nondoubling measures in ${\rm{R}}^n$ and applications, Trans. Amer. Math. Soc., 354 (2002), 2013-2033.  doi: 10.1090/S0002-9947-02-02922-7.  Google Scholar [14] N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theor. Biol., 79 (1979), 83-99.  doi: 10.1016/0022-5193(79)90258-3.  Google Scholar [15] E. M. Stein, Harmonic Analysis, Real Variable Methods, Orthogonality and Oscillatory Integrals, Princeton Univ. Press, Princeton, NJ, 1993. Google Scholar [16] P. Strzelecki, Gagliardo Nirenberg inequalities with a BMO term, Bull. London Math. Soc., 38 (2006), 294-300.  doi: 10.1112/S0024609306018169.  Google Scholar
 [1] Dung Le. On the regular set of BMO weak solutions to $p$-Laplacian strongly coupled nonregular elliptic systems. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3245-3265. doi: 10.3934/dcdsb.2014.19.3245 [2] Sun-Sig Byun, Lihe Wang. $W^{1,p}$ regularity for the conormal derivative problem with parabolic BMO nonlinearity in reifenberg domains. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 617-637. doi: 10.3934/dcds.2008.20.617 [3] Rafael De La Llave, R. Obaya. Regularity of the composition operator in spaces of Hölder functions. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 157-184. doi: 10.3934/dcds.1999.5.157 [4] Luca Lorenzi. Optimal Hölder regularity for nonautonomous Kolmogorov equations. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 169-191. doi: 10.3934/dcdss.2011.4.169 [5] Susanna Terracini, Gianmaria Verzini, Alessandro Zilio. Uniform Hölder regularity with small exponent in competition-fractional diffusion systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2669-2691. doi: 10.3934/dcds.2014.34.2669 [6] Maya Chhetri, D. D. Hai, R. Shivaji. On positive solutions for classes of p-Laplacian semipositone systems. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 1063-1071. doi: 10.3934/dcds.2003.9.1063 [7] Walter Allegretto, Yanping Lin, Shuqing Ma. Hölder continuous solutions of an obstacle thermistor problem. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 983-997. doi: 10.3934/dcdsb.2004.4.983 [8] Luciano Abadías, Carlos Lizama, Marina Murillo-Arcila. Hölder regularity for the Moore-Gibson-Thompson equation with infinite delay. Communications on Pure & Applied Analysis, 2018, 17 (1) : 243-265. doi: 10.3934/cpaa.2018015 [9] Yong Chen, Hongjun Gao, María J. Garrido–Atienza, Björn Schmalfuss. Pathwise solutions of SPDEs driven by Hölder-continuous integrators with exponent larger than $1/2$ and random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 79-98. doi: 10.3934/dcds.2014.34.79 [10] Samia Challal, Abdeslem Lyaghfouri. Hölder continuity of solutions to the $A$-Laplace equation involving measures. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1577-1583. doi: 10.3934/cpaa.2009.8.1577 [11] Lili Li, Chunrong Chen. Nonlinear scalarization with applications to Hölder continuity of approximate solutions. Numerical Algebra, Control & Optimization, 2014, 4 (4) : 295-307. doi: 10.3934/naco.2014.4.295 [12] Dung Le. Partial regularity of solutions to a class of strongly coupled degenerate parabolic systems. Conference Publications, 2005, 2005 (Special) : 576-586. doi: 10.3934/proc.2005.2005.576 [13] Zaiyun Peng, Xinmin Yang, Kok Lay Teo. On the Hölder continuity of approximate solution mappings to parametric weak generalized Ky Fan Inequality. Journal of Industrial & Management Optimization, 2015, 11 (2) : 549-562. doi: 10.3934/jimo.2015.11.549 [14] Charles Pugh, Michael Shub, Amie Wilkinson. Hölder foliations, revisited. Journal of Modern Dynamics, 2012, 6 (1) : 79-120. doi: 10.3934/jmd.2012.6.79 [15] Jinpeng An. Hölder stability of diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 315-329. doi: 10.3934/dcds.2009.24.315 [16] Angelo Favini, Rabah Labbas, Stéphane Maingot, Hiroki Tanabe, Atsushi Yagi. Necessary and sufficient conditions for maximal regularity in the study of elliptic differential equations in Hölder spaces. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 973-987. doi: 10.3934/dcds.2008.22.973 [17] Carlos Lizama, Luz Roncal. Hölder-Lebesgue regularity and almost periodicity for semidiscrete equations with a fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1365-1403. doi: 10.3934/dcds.2018056 [18] Jianhai Bao, Xing Huang, Chenggui Yuan. New regularity of kolmogorov equation and application on approximation of semi-linear spdes with Hölder continuous drifts. Communications on Pure & Applied Analysis, 2019, 18 (1) : 341-360. doi: 10.3934/cpaa.2019018 [19] Mark Pollicott. Local Hölder regularity of densities and Livsic theorems for non-uniformly hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2005, 13 (5) : 1247-1256. doi: 10.3934/dcds.2005.13.1247 [20] Luis Silvestre. Hölder continuity for integro-differential parabolic equations with polynomial growth respect to the gradient. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1069-1081. doi: 10.3934/dcds.2010.28.1069

2019 Impact Factor: 1.27