Qualitative Analysis Of Fixed Solution Problems For Some Partial Differential Equations

Partial differential equations are extensively applied to study different types of mathematical problems, such as physics, chemistry, and biology. The contents and methods of investigations are varied. In order to derive important theoretical and practical results the equations are supplemented with suitable additional conditions(usually, initial and boundary conditions). In this doctoral thesis we deal with some free boundary problems of reaction-diffusion equations and one Navier boundary problem of higher-order partial differential equation.Firstly, we investigate a free boundary problem with reaction terms vpand uq, which can be used as a model to describe heat propagation in a two-component combustible mixture. The main objective is to study the existence, uniqueness, regularity and long time behavior of the positive solution(maximal positive solution) to this problem. When p ≥ 1 and q ≥ 1, the local existence and uniqueness of solutions are established by applying the Lptheory for parabolic equations and the contraction mapping theorem, and then the unique solution is extended to the maximum existence interval in time. When p < 1 or q < 1, the existence and uniqueness of maximal positive solutions are proved by the method of approximation, and it is shown that the solution must blow up if the maximum existence time is finite. Then, the regularity of solutions is enhanced by means of the exterior Schauder estimate for parabolic equations, and the monotonicity of the free boundary is presented. Based upon the comparison principle, constructing proper upper and lower solutions generates some sufficient conditions for the blowup in finite time and the time-global positive solutions(maximal positive solutions). The long time behavior of bounded time-global solutions is investigated at last.Secondly, we explore two free boundary problems for the diffusive competition system in a higher space dimension with sign-changing coefficients. One may be viewed as describing how two competing species invade if they occupy an initial region; the other describes the dynamical process of a new competitor invading into the habitat of a native species. For the former problem, the time-global existence, uniqueness and estimates of the solutions are provided. Moreover, the sufficient conditions for spreading and vanishing of two species are given by introducing the corresponding eigenvalue problem, and then the dichotomy for spreading and vanishing is established. Furthermore, for the case of successful spreading, we manifest the long time behavior of the solution to this problem. For the latter, some similar results are achieved by modifying the arguments for the former; besides, a rough estimate for the spreading speed of free boundary is presented.Thirdly, we study a double-free-boundary problem for a nonlinear reaction-diffusion equation with different moving parameters. A local uniform convergence theorem is exhibited by employing ω-limit set, then some sufficient conditions are provided by constructing suitable upper and lower solutions, and then the sharp threshold for spreading and vanishing is established with the aid of continuity methods. Moreover, when spreading happens, we display the uniform convergence of solutions and the sharp estimates for free boundaries.Fourthly, we investigate a reaction-diffusion-advection equation with mixed and free boundary conditions. Our goal is to understand the effect of advective environment and no flux across the left boundary on the dynamics of this species. When the efficient of advection is small, we present the spreading-vanishing dichotomy and the threshold for spreading and vanishing, and show a much sharper estimate for the spreading speed of expanding front and the uniform convergence of solutions when spreading happens. In the final part we briefly describe the long time behavior of solutions for some cases of large advection.Finally, we discuss a Navier boundary problem of higher-order partial different equations in a upper half-space. The properties of solutions to this problem can be recognized by researching the corresponding integral equation with Bessel potential. We first enhance the regularity of positive solutions by regularity-lifting-method. Then, by employing the method of moving planes we demonstrate the symmetry of positive solutions, and then show that there is no positive solution for the integral equation.