Geodesic
Global and non-global existence of solutions to a nonlocal and degenerate quasilinear parabolic system
Czechoslovak Mathematical Journal, Tome 60 (2010) no. 3, pp. 675-688 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
Classification : 35D55, 35K05, 35K59, 35K65, 45K05
Keywords: strongly coupled; degenerate parabolic system; nonlocal source; global existence; blow-up 
@article{CMJ_2010_60_3_a6,
     author = {Chen, Yujuan},
     title = {Global and non-global existence of solutions to a nonlocal and degenerate quasilinear parabolic system},
     journal = {Czechoslovak Mathematical Journal},
     pages = {675--688},
     year = {2010},
     volume = {60},
     number = {3},
     mrnumber = {2672409},
     zbl = {1224.35157},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2010_60_3_a6/}
}
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Chen, Yujuan. Global and non-global existence of solutions to a nonlocal and degenerate quasilinear parabolic system. Czechoslovak Mathematical Journal, Tome 60 (2010) no. 3, pp. 675-688. http://geodesic.mathdoc.fr/item/CMJ_2010_60_3_a6/