In this paper we study the strong and weak property of travelling wave front solutions for a class of degenerate parabolic equations. How the strong and weak property changes under the effects of wave speed and reacti...In this paper we study the strong and weak property of travelling wave front solutions for a class of degenerate parabolic equations. How the strong and weak property changes under the effects of wave speed and reaction diffusion terms are showed.展开更多
In this paper we discuss the fundamental solution of the Keldysh type operator $ L_\alpha u \triangleq \frac{{\partial ^2 u}} {{\partial x^2 }} + y\frac{{\partial ^2 u}} {{\partial y^2 }} + \alpha \frac{{\partial u}} ...In this paper we discuss the fundamental solution of the Keldysh type operator $ L_\alpha u \triangleq \frac{{\partial ^2 u}} {{\partial x^2 }} + y\frac{{\partial ^2 u}} {{\partial y^2 }} + \alpha \frac{{\partial u}} {{\partial y}} $ , which is a basic mixed type operator different from the Tricomi operator. The fundamental solution of the Keldysh type operator with $ \alpha > - \frac{1} {2} $ is obtained. It is shown that the fundamental solution for such an operator generally has stronger singularity than that for the Tricomi operator. Particularly, the fundamental solution of the Keldysh type operator with $ \alpha < \frac{1} {2} $ has to be defined by using the finite part of divergent integrals in the theory of distributions.展开更多
A representation theorem for (x=±i0)λ lnk(x±i0) is proved and then the derivatives (lnk x±)', (xλ± lnk x±)', (x±(-n) x±1)', (d/dx){(x ± i0)λ lnk(x±i0)} and (...A representation theorem for (x=±i0)λ lnk(x±i0) is proved and then the derivatives (lnk x±)', (xλ± lnk x±)', (x±(-n) x±1)', (d/dx){(x ± i0)λ lnk(x±i0)} and (d/dx){(x±i0)-n lnk(x±i0)} are given.展开更多
文摘In this paper we study the strong and weak property of travelling wave front solutions for a class of degenerate parabolic equations. How the strong and weak property changes under the effects of wave speed and reaction diffusion terms are showed.
基金supported by the National Basic Research Program of China (Grant No.2006CB805902)National Natural Science Foundation of China (Grant No.10531020)the Research Foundation for Doctor Programme (Grant No.20050246001)
文摘In this paper we discuss the fundamental solution of the Keldysh type operator $ L_\alpha u \triangleq \frac{{\partial ^2 u}} {{\partial x^2 }} + y\frac{{\partial ^2 u}} {{\partial y^2 }} + \alpha \frac{{\partial u}} {{\partial y}} $ , which is a basic mixed type operator different from the Tricomi operator. The fundamental solution of the Keldysh type operator with $ \alpha > - \frac{1} {2} $ is obtained. It is shown that the fundamental solution for such an operator generally has stronger singularity than that for the Tricomi operator. Particularly, the fundamental solution of the Keldysh type operator with $ \alpha < \frac{1} {2} $ has to be defined by using the finite part of divergent integrals in the theory of distributions.
文摘A representation theorem for (x=±i0)λ lnk(x±i0) is proved and then the derivatives (lnk x±)', (xλ± lnk x±)', (x±(-n) x±1)', (d/dx){(x ± i0)λ lnk(x±i0)} and (d/dx){(x±i0)-n lnk(x±i0)} are given.
