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学术讲座: Entropy Condition and Very Singular Solution for Degenerate Diffusion Equation with Nonlinear Convection in One-Dimension
作者: 点击数:时间:2018-10-16 22:04:56

主讲人:卢国富教授

地 点:ope客户端下载_ope体育足彩师范教育学院微格录播教室录播厅

主办方:数学与信息工程学院

间:2018年10月20日下午16∶40~17∶20

讲座摘要:In this paper we consider the nonnegative singular solution of the following Cauchy problem:u_t=(u^m)_{xx}+(u^n)_x, (x, t)∈ R×(0, ∞), u(x, 0)=0, x≠ 0, and u(x,t) exhibits a singularity at the origin of (0,0), where the physics parameters m>1 and n> 0. In the previous works, the researches have been taken on so-called source-type solution. It is one type of singular solution to such problem. Here our attention is focused on the other type of singular solution of such problem. It is called very singular solution, which integral value of u(x,t) over R goes to +∞ as t→+0. This means very singular solution exhibits more singularity than that of source-type solution at (0,0). Different from the equation with absorption, the conservation law is confirmed here if the singular solution is smooth and decays enough fast as |x|→ ∞. Based on the previous works, we introduce the entropy condition and define the general solution of such equation to be less smooth than before. As a results, we overcome obstacles to the type of conservation law equation, and obtain the existence, uniqueness and nonexistence of very singular solution to such problem, i.e., we find two subcritical values of n_1(m) and n_2(m) only dependent of m such that if the convection in exponent range of n_2(m)<n<n_1(m) there is unique very singular solution, while if 0<n≤ n_2(m) or n≤ n_1(m) there is no very singular solution.

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