论文标题:一类拟线性吸收退化抛物方程的奇异解问题
Singular Solution of a Quasilinear Convection Diffusion Degenerate Parabolic Equation with Absorption
论文作者
论文导师 赵俊宁,论文学位 硕士,论文专业 基础数学
论文单位 厦门大学,点击次数 164,论文页数 31页File Size625K
2006-05-01论文网 http://www.lw23.com/lunwen_42463227/
P-Laplace equation; Cauchy Problem; Non-trivial solution
在这篇文章中我们讨论下面方程的初值问题的非负非平凡解的存在性问题 u_t=div(|▽u|~(p-2)▽u)-(?)/(?)x_ib_i(u)-u~q(x，t)∈S_T=R~N×(0，T) (1) u(x，0)=0 x∈R~N＼{0} (2) 其中p＞2，q＞0，b_i(s)∈C~1(R)。 我们证明了若p＞2成立，当0＜q＜p-1+p/N，0≤m＜p-1+p/N时，则方程(1)及初值为 u(x，0)=δ(x) x∈R~N (δ(x)是中心在原点的Dirac函数) (3)时存在弱解；当q＞p-1+p/N，0≤m≤q(p+N_p-N-1)/p+N_p-N时，问题(1)(3)没有解；当p-1＜q＜p-1+p/N，0≤m＜q时，问题(1)(2)有一个非常奇异解，即方程的一个解ω具有下列性质： ω∈C(S_T＼{(0，0)}) ω(x，0)=0 (?)x∈R~N＼{0} (?)∫_(|x|＜r)ω(x，t)dx=∞ (?)r＞0； 当q＞p-1+p/N，0＜m＜q-p/2N时，问题(1)(2)没有非常奇异解。在本文中所用的方法和文献[1]类似。
In this paper we discussed the Cauchy problem where p > 2, q > 0, b_i(s)∈C~1(R).In this paper, we are interested in the existence and nonexistence of non-negative and non-trivial solution of Cauchy problem (l)with initial datau(x,0) =δ(x) x∈R~N (3)where δ(x) denotes the Dirac mass centered at the origin.We have proved that let p > 2, if 0p-1 +p/N,0≤m≤q(p+Np-N-1)/p+Np-N , then(1)(3) has no solution; if p -1 < q < p -1+ p/N, 0≤ m < q, then (1)(2) has a very singular solution, i.e. a solution ω with the following properties:ω∈C(S|-_T\{(0,0)})ω(x,0) = 0 (?)x∈R~N\{0} then (1)(2) has no very singular solution. Here we use the methods similar to that in [1].
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