The global solution of a diffusion equation with nonlinear gradient term

被引:2
作者
Zhan, Huashui [1 ,2 ]
机构
[1] Xiamen Univ Technol, Sch Appl Math, Xiamen 361024, Peoples R China
[2] Jimei Univ, Sch Sci, Xiamen 361021, Peoples R China
来源
JOURNAL OF INEQUALITIES AND APPLICATIONS | 2013年
关键词
diffusion equation; Moser iteration; viscosity solution; extinction; PARABOLIC EQUATIONS; BEHAVIOR;
D O I
10.1186/1029-242X-2013-125
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
Consider the viscosity solution to the initial boundary value problem of the diffusion equation u(t) = div(vertical bar del u(m)vertical bar(p-2)del u(m)) - u(q1m)vertical bar del u(m)vertical bar(p1), with p > 1, m > 0, p(1) <= 2, p > 2p(1), its initial value u(x, 0) = u(0)(x) is an element of Lq-1+1/m (Omega), 3 > q > 1 and its boundary value u(x, t) = 0, (x, t) is an element of partial derivative Omega x (0, infinity). If p > 1+ 1/m, by considering the regularized problem and using Moser's iteration technique, we get the locally uniformly bounded property of the solution and the locally bounded property of the L-p-norm of the gradient. By the compactness theorem, the existence of the viscosity solution of the equation is obtained provided that mNq(1)/Nm(p - 1) - N + mq + p1(m(p - 1) + m - 2)/m(p - 1) - 1 < 1. If 2 < p < 1+ 1/m, the existence of solution is obtained in a similar way, and the extinction of the solution is proved in this case.
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页数:20
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