| Title:
|
Global and non-global existence of solutions to a nonlocal and degenerate quasilinear parabolic system (English) |
| Author:
|
Chen, Yujuan |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
60 |
| Issue:
|
3 |
| Year:
|
2010 |
| Pages:
|
675-688 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The paper deals with positive solutions of a nonlocal and degenerate quasilinear parabolic system not in divergence form $$ u_t = v^p\biggl (\Delta u + a\int _\Omega u \,{\rm d} x\biggr ),\quad v_t =u^q\biggl (\Delta v + b\int _\Omega v \,{\rm d} x\biggr ) $$ with null Dirichlet boundary conditions. By using the standard approximation method, we first give a series of fine a priori estimates for the solution of the corresponding approximate problem. Then using the diagonal method, we get the local existence and the bounds of the solution $(u,v)$ to this problem. Moreover, a necessary and sufficient condition for the non-global existence of the solution is obtained. Under some further conditions on the initial data, we get criteria for the finite time blow-up of the solution. (English) |
| Keyword:
|
strongly coupled |
| Keyword:
|
degenerate parabolic system |
| Keyword:
|
nonlocal source |
| Keyword:
|
global existence |
| Keyword:
|
blow-up |
| MSC:
|
35D55 |
| MSC:
|
35K05 |
| MSC:
|
35K59 |
| MSC:
|
35K65 |
| MSC:
|
45K05 |
| idZBL:
|
Zbl 1224.35157 |
| idMR:
|
MR2672409 |
| . |
| Date available:
|
2010-07-20T17:10:11Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140598 |
| . |
| Reference:
|
[1] Anderson, J. R., Deng, K.: Global existence for degenerate parabolic equations with a non-local forcing.Math. Methods Appl. Sci. 20 (1997), 1069-1087. Zbl 0883.35066, MR 1465394, 10.1002/(SICI)1099-1476(19970910)20:13<1069::AID-MMA867>3.0.CO;2-Y |
| Reference:
|
[2] Chen, H. W.: Analysis of blow-up for a nonlinear degenerate parabolic equation.J. Math. Anal. Appl. 192 (1995), 180-193. MR 1329419, 10.1006/jmaa.1995.1166 |
| Reference:
|
[3] Chen, Y., Gao, H.: Asymptotic blow-up behavior for a nonlocal degenerate parabolic equation.J. Math. Anal. Appl. 330 (2007), 852-863. Zbl 1113.35100, MR 2308412, 10.1016/j.jmaa.2006.08.014 |
| Reference:
|
[4] Deng, W., Li, Y., Xie, C.: Existence and nonexistence of global solutions of some nonlocal degenerate parabolic equations.Appl. Math. Lett. 16 (2003), 803-808. Zbl 1059.35066, MR 1986054, 10.1016/S0893-9659(03)80118-0 |
| Reference:
|
[5] Deng, W., Li, Y., Xie, C.: Global existence and nonexistence for a class of degenerate parabolic systems.Nonlinear Anal., Theory Methods Appl. 55 (2003), 233-244. Zbl 1032.35077, MR 2007471, 10.1016/S0362-546X(03)00226-8 |
| Reference:
|
[6] Duan, Z. W., Deng, W., Xie, C.: Uniform blow-up profile for a degenerate parabolic system with nonlocal source.Comput. Math. Appl. 47 (2004), 977-995. MR 2060331, 10.1016/S0898-1221(04)90081-8 |
| Reference:
|
[7] Duan, Z. W., Zhou, L.: Global and blow-up solutions for nonlinear degenerate parabolic systems with crosswise-diffusion.J. Math. Anal. Appl. 244 (2000), 263-278. Zbl 0959.35100, MR 1753038, 10.1006/jmaa.1999.6665 |
| Reference:
|
[8] Friedman, A., Mcleod, B.: Blow-up of positive solutions of semilinear heat equations.Indiana Univ. Math. J. 34 (1985), 425-447. Zbl 0576.35068, MR 0783924, 10.1512/iumj.1985.34.34025 |
| Reference:
|
[9] Friedman, A., Mcleod, B.: Blow-up of solutions of nonlinear degenerate parabolic equations.Arch. Ration. Mech. Appl. 96 (1987), 55-80. MR 0853975, 10.1007/BF00251413 |
| Reference:
|
[10] Gage, M. E.: On the size of the blow-up set for a quasilinear parabolic equation.Contemp. Math. 127 (1992), 41-58. Zbl 0770.35029, MR 1155408, 10.1090/conm/127/1155408 |
| Reference:
|
[11] Ladyzenskaya, O. A., Solonnikov, V. A., Ural'tseva, N. N.: Linear and Quasilinear Equations of Parabolic Type.American Mathematical Society Providence (1968). |
| Reference:
|
[12] Pao, C. V.: Nonlinear Parabolic and Elliptic Equations.Plenum Press New York (1992). Zbl 0777.35001, MR 1212084 |
| Reference:
|
[13] Passo, R. Dal, Luckhaus, S.: A degenerate diffusion problem not in divergence form.J. Differ. Equations 69 (1987), 1-14. MR 0897437, 10.1016/0022-0396(87)90099-4 |
| Reference:
|
[14] Wang, M. X.: Some degenerate and quasilinear parabolic systems not in divergence form.J. Math. Anal. Appl. 274 (2002), 424-436. Zbl 1121.35321, MR 1936706, 10.1016/S0022-247X(02)00347-5 |
| Reference:
|
[15] Wang, M. X., Xie, C. H.: A degenerate and strongly coupled quasilinear parabolic system not in divergence form.Z. Angew. Math. Phys. 55 (2004), 741-755. Zbl 1181.35132, MR 2087763, 10.1007/s00033-004-1133-4 |
| Reference:
|
[16] Wang, S., Wang, M. X., Xie, C. H.: A nonlinear degenerate diffusion equation not in divergence form.Z. Angew. Math. Phys. 51 (2000), 149-159. Zbl 0961.35077, MR 1745296, 10.1007/PL00001503 |
| Reference:
|
[17] Wiegner, M.: A degenerate diffusion equation with a nonlinear source term.Nonlinear Anal., Theory Methods Appl. 28 (1997), 1977-1995. Zbl 0874.35061, MR 1436366, 10.1016/S0362-546X(96)00027-2 |
| Reference:
|
[18] Zimmer, T.: On a degenerate parabolic equation. IWR Heidelberg.Preprint 93-05 (1993). |
| . |
