On the interfaces in a nonlocal quasilinear degenerate equation arising in population dynamics
On the interfaces in a nonlocal quasilinear degenerate equation arising in population dynamics
复制标题
关于种群动态中出现的非局部拟线性简并方程中的接口
DOI：
10.1007/bf03167255
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发表时间：
1996
影响因子：
0.9
通讯作者：
S. Shmarev
中科院分区：
文献类型：
--
作者：
J. I. Díaz;T. Nagai;S. Shmarev
本文研究了方程Cauchy问题非负弱解存在区域的分界面的正则性和传播性质
$$u_t =（u^m）xx + \left[ {u\left（{\int_{ - \infty }^x {u（y，t）dy} - \int_x^\infty {u（y，t）dy} } \right）} \right]_x，m > 1，$$
严格为正或等于零。结果表明，在适当的初始条件下，界面是（不一定是单调的）C∞曲线，而且在转折点处不失去这种规律性。界面规则性的研究是通过拉格朗日坐标。我们证明了界面的初始性态是由初始支撑端点附近的初始基准面的增长特性决定的。估计（从下面和从上面）的正性集的解决方案的宽度也得到了。
AbstractWe study regularity and propagation properties of interfaces separating regions where nonnegative weak solutions of the Cauchy problem for the equation
$$u_t = (u^m )xx + \left[ {u\left( {\int_{ - \infty }^x {u(y,t)dy} - \int_x^\infty {u(y,t)dy} } \right)} \right]_x , m > 1,$$
are strictly positive or equal to zero. It is shown that under suitable conditions on the initial data the interfaces are (not necessarily monotone)C∞-curves and they do not lose this regularity at their turning points. The study of the interface regularity is performed via Lagrangian coordinates. We show that the initial behaviorof interfaces is determined by the character of growth of the initial datum near the endpoints of the initial support. Estimates (from below and from above) on the width of the positivity set of solutions are also obtained.