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Evolution of interfaces for the nonlinear parabolic p-Laplacian-type reaction-diffusion equations. II. Fast diffusion vs. absorption
European Journal of Applied Mathematics ( IF 1.9 ) Pub Date : 2019-03-18 , DOI: 10.1017/s095679251900007x
UGUR G. ABDULLA , ROQIA JELI

We present a full classification of the short-time behaviour of the interfaces and local solutions to the nonlinear parabolic p-Laplacian-type reaction-diffusion equation of non-Newtonian elastic filtration$$u_t-\Big(|u_x|^{p-2}u_x\Big)_x+bu^{\beta}=0, \ 1 \lt p \lt 2, \beta \gt 0.$$If the interface is finite, it may expand, shrink or remain stationary as a result of the competition of the diffusion and reaction terms near the interface, expressed in terms of the parameters p, β, sign b, and asymptotics of the initial function near its support. In some range of parameters, strong domination of the diffusion causes infinite speed of propagation and interfaces are absent. In all cases with finite interfaces, we prove the explicit formula for the interface and the local solution with accuracy up to constant coefficients. We prove explicit asymptotics of the local solution at infinity in all cases with infinite speed of propagation. The methods of the proof are based on nonlinear scaling laws and a barrier technique using special comparison theorems in irregular domains with characteristic boundary curves. A full description of small-time behaviour of the interfaces and local solutions near the interfaces for slow diffusion case when p>2 is presented in a recent paper by Abdulla and Jeli [(2017) Europ. J. Appl. Math.28(5), 827–853].

中文翻译:

非线性抛物线 p-拉普拉斯型反应扩散方程的界面演化。二、快速扩散与吸收

我们提出了界面的短时行为的完整分类和非线性抛物线的局部解p-非牛顿弹性过滤的拉普拉斯型反应-扩散方程$$u_t-\Big(|u_x|^{p-2}u_x\Big)_x+bu^{\beta}=0, \ 1 \lt p \lt 2, \beta \gt 0.$$如果界面是有限的,由于界面附近的扩散和反应项的竞争,它可能会膨胀、收缩或保持静止,用参数表示p,β, 标志b,以及在其支持附近的初始函数的渐近线。在某些参数范围内,扩散的强支配导致无限的传播速度并且不存在界面。在所有有限界面的情况下,我们证明了界面的显式公式和局部解,精度高达恒定系数。我们证明了在无限传播速度的所有情况下,局部解的显式渐近性。证明方法基于非线性标度律和障碍技术,在具有特征边界曲线的不规则域中使用特殊比较定理。慢扩散情况下界面的小时间行为和界面附近的局部解的完整描述p>2 由 Abdulla 和 Jeli 在最近的一篇论文中提出 [(2017)欧洲。J.应用。数学。28(5), 827–853]。
更新日期:2019-03-18
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