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Volume 32, Issue 3
Parabolic System Related to the P-Laplician with Degeneracy on the Boundary

Qitong Ou & Huashui Zhan

DOI: 10.4208/jpde.v32.n3.5

J. Part. Diff. Eq., 32 (2019), pp. 281-292.

Published online: 2019-10

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  • Abstract

In this article, we study the system with boundary degeneracy

$u_{it}-{\rm div}(a(x)|\triangledown u_{i}|^{p_{i}-2}\nabla u_i)=f_{i}(x,t,u_1,u_2),\qquad (x,t)\in\Omega_T$.

Applying the monotone iterattion technique and the regularization method, we get the existence of solution for a regularized system. Moreover, under an integral condition on the coefficient function $a(x)$, %  And if  %$ \int_{\Omega}  a(x)^{-\frac{1}{min{(p_1,p_2)}-1}} {\rm d}x{\rm d}t\leq C ,$ the existence and the uniqueness of the local solutions of the system is obtained by using a standard limiting process. Finally, the stability of the solutions is proved without any boundary value condition, provided $a(x)$ satisfies another restriction.

  • Keywords

Weak solution boundary degeneracy parabolic system initial boundary value problem existence, stability.

  • AMS Subject Headings

35B40, 35K65, 35K55

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

ouqitong@xmut.edu.cn (Qitong Ou)

2012111007@xmut.edu.cn (Huashui Zhan)

  • BibTex
  • RIS
  • TXT
@Article{JPDE-32-281, author = {Ou , Qitong and Zhan , Huashui}, title = {Parabolic System Related to the P-Laplician with Degeneracy on the Boundary}, journal = {Journal of Partial Differential Equations}, year = {2019}, volume = {32}, number = {3}, pages = {281--292}, abstract = {

In this article, we study the system with boundary degeneracy

$u_{it}-{\rm div}(a(x)|\triangledown u_{i}|^{p_{i}-2}\nabla u_i)=f_{i}(x,t,u_1,u_2),\qquad (x,t)\in\Omega_T$.

Applying the monotone iterattion technique and the regularization method, we get the existence of solution for a regularized system. Moreover, under an integral condition on the coefficient function $a(x)$, %  And if  %$ \int_{\Omega}  a(x)^{-\frac{1}{min{(p_1,p_2)}-1}} {\rm d}x{\rm d}t\leq C ,$ the existence and the uniqueness of the local solutions of the system is obtained by using a standard limiting process. Finally, the stability of the solutions is proved without any boundary value condition, provided $a(x)$ satisfies another restriction.

}, issn = {2079-732X}, doi = {https://doi.org/10.4208/jpde.v32.n3.5}, url = {http://global-sci.org/intro/article_detail/jpde/13343.html} }
TY - JOUR T1 - Parabolic System Related to the P-Laplician with Degeneracy on the Boundary AU - Ou , Qitong AU - Zhan , Huashui JO - Journal of Partial Differential Equations VL - 3 SP - 281 EP - 292 PY - 2019 DA - 2019/10 SN - 32 DO - http://doi.org/10.4208/jpde.v32.n3.5 UR - https://global-sci.org/intro/article_detail/jpde/13343.html KW - Weak solution KW - boundary degeneracy parabolic system KW - initial boundary value problem KW - existence, stability. AB -

In this article, we study the system with boundary degeneracy

$u_{it}-{\rm div}(a(x)|\triangledown u_{i}|^{p_{i}-2}\nabla u_i)=f_{i}(x,t,u_1,u_2),\qquad (x,t)\in\Omega_T$.

Applying the monotone iterattion technique and the regularization method, we get the existence of solution for a regularized system. Moreover, under an integral condition on the coefficient function $a(x)$, %  And if  %$ \int_{\Omega}  a(x)^{-\frac{1}{min{(p_1,p_2)}-1}} {\rm d}x{\rm d}t\leq C ,$ the existence and the uniqueness of the local solutions of the system is obtained by using a standard limiting process. Finally, the stability of the solutions is proved without any boundary value condition, provided $a(x)$ satisfies another restriction.

Ou , Qitong and Zhan , Huashui. (2019). Parabolic System Related to the P-Laplician with Degeneracy on the Boundary. Journal of Partial Differential Equations. 32 (3). 281-292. doi:10.4208/jpde.v32.n3.5
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