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 and 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.

[2]

H. Amann, ynamic theory of quasilinear parabolic equations Ⅱ. Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75. 

[3]

H. Amann, Dynamic theory of quasilinear parabolic systems Ⅲ. Global existence, Math Z., 202 (1989), 219-250.  doi: 10.1007/BF01215256.

[4]

B. FranchiC. 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.

[5]

A. Friedman, Partial Differential Equations, New York, 1969.

[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.

[7]

E. Giusti, Direct Methods in the Calculus of Variations World Scientific, 2003.

[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.

[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.

[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.

[11]

D. LeL. Nguyen and T. Nguyen, Coexistence in Cross Diffusion systems, Indiana Univ. J. Math., 56 (2007), 1749-1791. 

[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.

[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.

[14]

N. ShigesadaK. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theor. Biol., 79 (1979), 83-99.  doi: 10.1016/0022-5193(79)90258-3.

[15]

E. M. Stein, Harmonic Analysis, Real Variable Methods, Orthogonality and Oscillatory Integrals, Princeton Univ. Press, Princeton, NJ, 1993.

[16]

P. Strzelecki, Gagliardo Nirenberg inequalities with a BMO term, Bull. London Math. Soc., 38 (2006), 294-300.  doi: 10.1112/S0024609306018169.

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.

[2]

H. Amann, ynamic theory of quasilinear parabolic equations Ⅱ. Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75. 

[3]

H. Amann, Dynamic theory of quasilinear parabolic systems Ⅲ. Global existence, Math Z., 202 (1989), 219-250.  doi: 10.1007/BF01215256.

[4]

B. FranchiC. 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.

[5]

A. Friedman, Partial Differential Equations, New York, 1969.

[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.

[7]

E. Giusti, Direct Methods in the Calculus of Variations World Scientific, 2003.

[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.

[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.

[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.

[11]

D. LeL. Nguyen and T. Nguyen, Coexistence in Cross Diffusion systems, Indiana Univ. J. Math., 56 (2007), 1749-1791. 

[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.

[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.

[14]

N. ShigesadaK. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theor. Biol., 79 (1979), 83-99.  doi: 10.1016/0022-5193(79)90258-3.

[15]

E. M. Stein, Harmonic Analysis, Real Variable Methods, Orthogonality and Oscillatory Integrals, Princeton Univ. Press, Princeton, NJ, 1993.

[16]

P. Strzelecki, Gagliardo Nirenberg inequalities with a BMO term, Bull. London Math. Soc., 38 (2006), 294-300.  doi: 10.1112/S0024609306018169.

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