Global and non-global existence of solutions to a nonlocal and degenerate quasilinear parabolic system

Yujuan Chen

Czechoslovak Mathematical Journal (2010)

  • Volume: 60, Issue: 3, page 675-688
  • ISSN: 0011-4642

Abstract

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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 Δ u + a ∫ Ω u d x , v t = u q Δ v + b ∫ Ω v d x 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.

How to cite

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Chen, Yujuan. "Global and non-global existence of solutions to a nonlocal and degenerate quasilinear parabolic system." Czechoslovak Mathematical Journal 60.3 (2010): 675-688. <http://eudml.org/doc/38035>.

@article{Chen2010,
abstract = {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.},
author = {Chen, Yujuan},
journal = {Czechoslovak Mathematical Journal},
keywords = {strongly coupled; degenerate parabolic system; nonlocal source; global existence; blow-up; strongly coupled; degenerate parabolic system; nonlocal source; global existence; blow-up},
language = {eng},
number = {3},
pages = {675-688},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Global and non-global existence of solutions to a nonlocal and degenerate quasilinear parabolic system},
url = {http://eudml.org/doc/38035},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Chen, Yujuan
TI - Global and non-global existence of solutions to a nonlocal and degenerate quasilinear parabolic system
JO - Czechoslovak Mathematical Journal
PY - 2010
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 60
IS - 3
SP - 675
EP - 688
AB - 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.
LA - eng
KW - strongly coupled; degenerate parabolic system; nonlocal source; global existence; blow-up; strongly coupled; degenerate parabolic system; nonlocal source; global existence; blow-up
UR - http://eudml.org/doc/38035
ER -

References

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