- 作者：Nguyen Thanh Long ; Vo Giang Giai
- 关键词：Faedo– ; Galerkin method ; Existence and uniqueness of a weak solution ; Energy-type estimates ; Compactness ; Asymptotic expansion
- 刊名：Nonlinear Analysis
- 出版年：2007
- 出版时间：15 September 2007
- 年：2007
- 卷：67
- 期：6
- 页码：1791-1819
- 全文大小：477 K
In this paper, we consider three main parts. In Part 1, under the conditions , , μ(x,t)≥μ0>0, μtL1(0,T;L∞), μt(x,t)≤0, a.e. (x,t)QT; K0, K1≥0; p0, q0, p1, q1≥2, , the function f supposed to be continuous with respect to two variables and nondecreasing with respect to the second variable and some others, we prove that the problem (1) and (2) has a weak solution (u,P). If, in addition, 32b15b8871f473313da3824d75209733"" title=""Click to view the MathML source"">kW1,1(0,T), p0, p1{2}[3,+∞) and some other conditions, then the solution is unique. The proof is based on the Faedo–Galerkin method and the weak compact method associated with a monotone operator. For the case of q0=q1=2;p0,p1≥2, in Part 2 we prove that the unique solution (u,P) belongs to (L∞(0,T;H2)∩C0(0,T;H1)∩C1(0,T;L2))×H1(0,T), with utL∞(0,T;H1), uttL∞(0,T;L2), u(0,), u(1,)H2(0,T), if we assume , and some other conditions. Finally, in Part 3, with q0=q1=2; p0, p1≥N+1, , N≥2, we obtain an asymptotic expansion of the solution (u,P) of the problem (1) and (2) up to order N+1 in two small parameters K0, K1.