| Nonlinear parabolic equation is usually called reaction-diffusion equation, whose diffusion operator is classical Laplace operator. However, the non-local diffusion op-erator described by convolution also has a deep research background. The traveling wave solution and entire solution, as special solutions of diffusion equations, can be good descriptions of several practical problems. This paper aims to study the traveling wave solution and entire solutions of an epidemics model by oral-fecal transmission with nonlocal diffusion.In the first place, this paper investigates the existence of traveling wave solution by the means of upper solution and lower solution construction just like Pan et al. Considering the characteristic equation of the system, we give out a pair of upper and lower solutions using the exponential function associated with smallest eigenvalue and prove the existence of traveling wave solutions. Different from classical diffusion system, the characteristic function of nonlocal diffusion system is the product of two integral terms.In addition, we further research the uniqueness and asymptotic behavior of traveling wave solution of the system. Estimation of the traveling wave solution in negative infinity and Ikehara’s theorem are used to depict its asymptotic behavior. Meanwhile, when the speed is less than the minimum speed, the nonexistence of the traveling wave solution has been proven. On the basis of asymptotic behavior, the uniqueness of traveling wave solution is proved by the calculation of sliding planes.Finally, the entire solutions of the system are considered. The existence of spatially independent solution, Cauchy problem of the system and the comparison principle are established. By considering the interaction of two traveling wave fronts propagating from both directions of the x-axis with different speeds and a spatially independent solution, three new types of entire solutions are constructed. We also gain their some properties. |
PDF Full Text Request