Existence of regular solutions to a class of parabolic systems in two space dimensions with critical growth behaviour

Annali dell'Universita di Ferrara

Volumn 55, Issue 2, 2009, Pages 239-261

  Frehse, Jens ^a   Specovius Neugebauer, Maria ^b  

^a UNIVERSITY OF BONN   (Germany)

^b UNIVERSITY OF KASSEL   (Germany)

Author keywords

Global solutions; H lder continuity; Nonlinear parabolic systems

Indexed keywords

EID: 84878381108     PISSN: 04303202     EISSN: 18271510     Source Type: Journal    
DOI: 10.1007/s11565-009-0071-7     Document Type: Article

References (20)

1. Amann, H.: Quasilinear parabolic systems under nonlinear boundary conditions. Arch. Ration. Mech. Anal. 92, 153-192 (1986).
2. Arkhipova, A.: Cauchy-Neumann problem for a class of nondiagonal parabolic systems with quadratic growth nonlinearities. I. On the continuability of smooth solutions. Comment. Math. Univ. Carolin. 41, 693-718 (2000).
3. Arkhipova, A.: Continuability in time of smooth solutions of strong-nonlinear nondiagonal parabolic systems. Ann. Sc. Norm. Super. Pisa Cl. Sci. 1(5), 153-167 (2002).
4. Arkhipova, A. A.: Time-local and time global solvability of the Cauchy-Dirichlet problem for a class of nonlinear non-diagonal parabolic systems. Algebra i Analiz 11, 69-102 (1999) (translation in St. Petersburg Math. J. 11(6), 989-1017, 2000).
5. Bensoussan, A., Frehse, J.: Smooth solutions of systems of quasilinear parabolic equations. ESAIM Control Optim. Calc. Var. 8, 169-193 (electronic). A tribute to J. L. Lions (2002).
6. Eells, J. Jr., Sampson, J. H.: Harmonic mappings of Riemannian manifolds. Am. J. Math. 86, 109-160 (1964).
7. Fattorusso, L.: Differentiability of weak solutions of nonlinear second order parabolic systems with quadratic growth and nonlinearity q ≥ 2. Comment. Math. Univ. Carolin. 45, 73-90 (2004).
8. Frehse, J.: An existence theorem for a class of non-coercive optimization and variational problems. Math. Z. 159, 51-63 (1978).
9. Frehse, J., Specovius-Neugebauer, M.: Morrey estimates and Hölder continuity for solutions to parabolic equations with entropy inequalities. J. Reine Angew. Math. (2009, to appear).
10. Frehse, J., Specovius-Neugebauer, M.: Existence of Hölder continuous Young measure solutions to coercive non-monotone parabolic systems in two space dimensions. Preprint no. 441 of SFB 611. University of Bonn, Bonn (2009).
11. Giaquinta, M., Modica, G.: Local existence for quasilinear parabolic systems under nonlinear boundary conditions. Ann. Mat. Pura Appl. 149(4), 41-59 (1987).
12. Gilbarg, D., Trudinger, N. S.: Elliptic Partial Differential Equations of Second Order. Grundlehren der Mathematischen Wissenschaften. Springer, Berlin (1983).
13. Hamilton, R. S.: Harmonic Maps of Manifolds With Boundary. Lecture Notes in Mathematics, vol. 471. Springer, Berlin (1975).
14. Ladyženskaja, O. A., Solonnikov, V. A., Ural'ceva, N. N.: Linear andQuasilinear Equations of Parabolic Type. Translated from the Russian by S. Smith. Translations of Mathematical Monographs, vol. 23, American Mathematical Society, Providence, R. I. (1967).
15. Morrey, C. B. Jr.: Multiple Integrals in the Calculus of Variations. Springer, Berlin (1966).
16. Marino, M., Maugeri, A.: Differentiability of weak solutions of nonlinear parabolic systems with quadratic growth. Matematiche (Catania) 50, 361-377 (1996).
17. Marino, M., Maugeri, A.: A remark on the note: partial Hölder continuity of the spatial derivatives of the solutions to nonlinear parabolic systems with quadratic growth. Rend. Sem. Mat. Univ. Padova 95, 23-28 (1996).
18. Naumann, J.: Everywhere Hölder continuity of the spatial gradient of weak solutions to nonlinear parabolic systems. Rend. Circ. Mat. Palermo. 47(2), 409-430 (1998).
19. Naumann, J., Wolf, J.: Interior differentiability of weak solutions to parabolic systems with quadratic growth nonlinearities. Rend. Sem. Mat. Univ. Padova 98, 253-272 (1997).
20. Struwe, M.: On the evolution of harmonic mappings of Riemannian surfaces. Comment. Math. Helv. 60, 558-581 (1985).