Spreading speed and sharp asymptotic profiles of solutions in free boundary problems for nonlinear advection-diffusion equations
Spreading speed and sharp asymptotic profiles of solutions in free boundary problems for nonlinear advection-diffusion equations
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DOI：
10.1016/j.jmaa.2015.02.051
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发表时间：
2015-08
影响因子：
1.3
通讯作者：
Y. Kaneko;H. Matsuzawa
中科院分区：
文献类型：
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作者：
Y. Kaneko;H. Matsuzawa
本文考虑非线性对流扩散方程ut − uxx + β ux = f（u）（t> 0，g（t）< x< h（t））的自由边界问题，其中x= g（t）和x= h（t）是自由边界.这个问题可以用来描述生物或化学物种的扩散，其中自由边界代表扩展的前沿。当f为逻辑非线性时，由于平流项的存在，两个波前h（t）和g（t）的渐近传播速度是不同的。在本文中，我们对单稳态、双稳态和燃烧型非线性方程，给出了波前传播速度的精确估计，并证明了当传播发生时，解在C2范数下收敛于半波.我们开发了新的方法，并延长了以前的结果与平流项的问题。
In this study, we consider free boundary problems for nonlinear advection–diffusion equations of the form u t− u x x+ β u x= f (u) for t> 0, g (t)< x< h (t), where x= g (t) and x= h (t) are free boundaries. This problem may be used to describe the spreading of a biological or chemical species where the free boundaries represent the expanding fronts. When f is a logistic nonlinearity, it has been shown that the asymptotic spreading speeds of the two fronts h (t) and g (t) are different due to the advection term. In this study, for monostable, bistable, and combustion types nonlinearities, we give much sharper estimates of the different spreading speeds of the fronts, and we also prove that the solution converges to a semi-wave in C 2-norm as t→∞ when spreading occurs. We develop new approaches and extend a previous result to the problem with the advection term.