Qualitative Properties Of Solutions For A Class Of Advection Equations With Combustion Nonlinearity

In this paper,the free boundary problem of reaction-convection-diffusion equation with a combustion nonlinear term is studied.The asymptotic behavior of the solution under different convection intensities is considered.Firstly,the equilibrium solutions are classified by the method of phase-plane analysis,and the intensity of convection is divided into two cases:small convection and large convection.Under these two kinds of convection intensity,the problem has completely different equilibrium solution classification.For the case of large convection,we construct a suitable upper solution to obtain the solution uniformly converges to 0 in the limit interval I_∞.In the case of small convection,we use the ω-limit set and the property of zero point to obtain the local uniform convergence to0 or 1.Then,for a family of continuously varying initial values,we discuss the convergence of the solution in the case of small convection.By using the method of upper and lower solutions,we obtain that when the initial value is small,I_∞is a bounded interval,and the solution uniformly converges to 0 in I_∞.When the initial value is sufficiently large,the solution uniformly converges to 1 and I_∞.When the initial value is a critical initial value,neither of the above-mentioned situations will occur.Finally,we discuss the convergence of the solution in the critical state.In this paper,we give some important traveling wave solutions,compare them with the true solutions,and discuss the change of the number of intersection points by using the property of zero point.Ultimately,the convergence of the critical solution in the case of small convection is obtained.In particular,the monotonic half-wave does not only exist in small convection cases,so the intensity of convection is subdivided into small,medium and large convection cases.We obtain the existence of non-monotonic half-wave if and only if in small and intermediate convection cases.