Parabolic System Related to the P-Laplician with Degeneracy on the Boundary

Abstract

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In this article, we study the system with boundary degeneracy<\/p>

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

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 $a(x)$, %\u00a0 And if\u00a0 %$ \\int_{\\Omega}\u00a0 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.<\/p>"