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Keywords:
strongly coupled; degenerate parabolic system; nonlocal source; global existence; blow-up
strongly coupled; degenerate parabolic system; nonlocal source; global existence; blow-up
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.
References:
[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. DOI 10.1002/(SICI)1099-1476(19970910)20:13<1069::AID-MMA867>3.0.CO;2-Y | MR 1465394 | Zbl 0883.35066
[2] Chen, H. W.: Analysis of blow-up for a nonlinear degenerate parabolic equation. J. Math. Anal. Appl. 192 (1995), 180-193. DOI 10.1006/jmaa.1995.1166 | MR 1329419
[3] Chen, Y., Gao, H.: Asymptotic blow-up behavior for a nonlocal degenerate parabolic equation. J. Math. Anal. Appl. 330 (2007), 852-863. DOI 10.1016/j.jmaa.2006.08.014 | MR 2308412 | Zbl 1113.35100
[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. DOI 10.1016/S0893-9659(03)80118-0 | MR 1986054 | Zbl 1059.35066
[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. DOI 10.1016/S0362-546X(03)00226-8 | MR 2007471 | Zbl 1032.35077
[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. DOI 10.1016/S0898-1221(04)90081-8 | MR 2060331
[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. DOI 10.1006/jmaa.1999.6665 | MR 1753038 | Zbl 0959.35100
[8] Friedman, A., Mcleod, B.: Blow-up of positive solutions of semilinear heat equations. Indiana Univ. Math. J. 34 (1985), 425-447. DOI 10.1512/iumj.1985.34.34025 | MR 0783924 | Zbl 0576.35068
[9] Friedman, A., Mcleod, B.: Blow-up of solutions of nonlinear degenerate parabolic equations. Arch. Ration. Mech. Appl. 96 (1987), 55-80. DOI 10.1007/BF00251413 | MR 0853975
[10] Gage, M. E.: On the size of the blow-up set for a quasilinear parabolic equation. Contemp. Math. 127 (1992), 41-58. DOI 10.1090/conm/127/1155408 | MR 1155408 | Zbl 0770.35029
[11] Ladyzenskaya, O. A., Solonnikov, V. A., Ural'tseva, N. N.: Linear and Quasilinear Equations of Parabolic Type. American Mathematical Society Providence (1968).
[12] Pao, C. V.: Nonlinear Parabolic and Elliptic Equations. Plenum Press New York (1992). MR 1212084 | Zbl 0777.35001
[13] Passo, R. Dal, Luckhaus, S.: A degenerate diffusion problem not in divergence form. J. Differ. Equations 69 (1987), 1-14. DOI 10.1016/0022-0396(87)90099-4 | MR 0897437
[14] Wang, M. X.: Some degenerate and quasilinear parabolic systems not in divergence form. J. Math. Anal. Appl. 274 (2002), 424-436. DOI 10.1016/S0022-247X(02)00347-5 | MR 1936706 | Zbl 1121.35321
[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. DOI 10.1007/s00033-004-1133-4 | MR 2087763 | Zbl 1181.35132
[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. DOI 10.1007/PL00001503 | MR 1745296 | Zbl 0961.35077
[17] Wiegner, M.: A degenerate diffusion equation with a nonlinear source term. Nonlinear Anal., Theory Methods Appl. 28 (1997), 1977-1995. DOI 10.1016/S0362-546X(96)00027-2 | MR 1436366 | Zbl 0874.35061
[18] Zimmer, T.: On a degenerate parabolic equation. IWR Heidelberg. Preprint 93-05 (1993).
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