The Studies On Behavior Of Solutions To A Non-Newtonian Polytropic Filtration Equation

Non-Newtonian polytropic filtration equation comes from a variety of diffusion phenomena appeared widely in nature. They are suggested as mathematical models of physical problems in many fields such as filtration, phasetransition, biochemistry and dynamics of biological groups. We mainly investigate the global and blow-up of the positive solutions to the non-Newtonian polytropic filtration equation with nonlocal source subject to weighted nonlocal Dirichlet boundary conditions and the critical exponents of the equation with multiple nonlinearities.In Chapter One, we mianly introduce the physical background, history and development of the equation that we consider.In Chapter Two, we consider the following non-Newton polytropic filtration equation with nonlocal source subject to weighted nonlocal Dirichlet boundary conditions (?)and discuss the influence of the weighted function to the global existence and blow-up of the solutions. Owing to the doubly degeneracy of the equation, we first define a weak solution of the problem and give a local existence result of the weak solution. Then we prove the weak comparison principle by constructing the test function and applying the Gronwall's inequality. Finally, we obtain the result of the global existence and blow-up of the solution by constructing all kinds of upper-lower solutions.In Chapter Three, we consider the following non-Newton polytropic filtration equation with a source and nonlinear boundary flux (?)Appling the weak comparison principle and upper-lower methods, we establish the sufficient conditions for the global existence and blow-up of the solution by constructing various self-similar supersolutions and subsolutions and then obtain the critical global existence curve and critical Fujita curve.