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This paper mainly describes and summarizes the research works on Non-Newtonian polytropic filtration equations in recent years. In the first part, we summarize simply some properties of the solutions of the Non-Newtonian polytropic filtration equations, and in the other part, we summarize some new results for the Non-Newtonian polytropic filtration equations.This paper summarizes the existence and uniqueness of the solutions of the Non-Newtonian filtration equations.We consider for the follow problemWe summarize some properties of the solutions about the problem（ⅰ）（ⅱ）. In one dimension case, we consider the problem Then we haveTheorem Assume m, p >0, mp >1, u0∈L1 （ R）,then there exists a function u satisfies the following: （e） If u （ x , t ） ,（u|^） （ x ,t ） are the solutions of u0（x） ,（u|^）0（x） ,then Specially for u0≤（u|^）0, u （ x , t ）≤（u|^） （ x ,t ）and t >0, x∈R. If u is the Barenblatt solution,then C （ m , p ）> 0 is better.（g） If u0∈L1 （R）∩L∞R,then（j） For t∈(τ,T ],0<τ<∞In higher dimension case. We consider the problem: ThenTheorem （1）If u0∈Lr （Ω）,β+ 1≤r≤∞, then there exists a u is a warm solution of（1）—（3）,satisfies （2）If , then there exist a u is a warm solution of（1）—（3）,satisfies and（3）Ifβ< P-1≤γ（λ>0 ） orβ< P-1 （λ=0 ）,then there exist a u is a strong solution for the problem（ⅰ）,（ⅱ）,satisfies the following: If then（4）Ifβ≥1, then a u is a weak solution of（ⅰ）,（ⅱ）satisfies: If thenTheorem Assume u,v are warm solutions of（1）—（3）with the same initial value,then Theorem Assume m >0, p >1, m （ p- 1 ） >1, u0≥0, u0∈L1 （ RN）, then there exists a u is a weak solution of（ⅰ）,（ⅱ）, satisfiesTheorem Assume u and v are the solutions of（ⅰ）,（ⅱ）with initial value,thenNext we describe the other properties of the solutions. The long time of the solution:For the Barenblatt solution of （ⅰ）,（ⅱ） Ec is the solution of the problem（ⅰ）,（ⅱ）withTheorem Assume m （ p- 1 ） >1, u0∈L1 （ RN）.If E c is the only solution othe problem（ⅰ）,（a）, then in And u is the solution of the problem（ⅰ）,（ⅱ）. The boundedness estimate of the solutionWe consider the problem: Then we have Theorem Assumethen for （?）t >0,there exists C （ m,d,p,q0 ）, satisfies andExponent for the Non-Newtonian polytropic filtration equations with nonlinear boundary conditionsWe consider the problem: We haveExtinction and positivityWe consider the problem:Theorem If u is a weak solution,and 1< p< 1+ 1/m,then exist finite time T ,satisfiesTheorem If u is a weak solution,and p≥1+1/m, then exist finite time T ,satisfiesExistence of solutions to a class of degenerate parabolic equations in Non-divergence formWe discuss the existence of weak solutions to the initial and boundary value problem of a class of degenerate parabolic equations in non-divergence form. The authors applied the method of parabolic regularization to prove the existence of weak solutions to the problem.We consider the problem Theorem Assume p≥2,γ∈（0,1）, then there exists a weak solution andBlow-up and boundednessWe discuss the blow-up and boundedness of solutions at finite time for evolution p-Laplace equations with nonlinear boundary conditions. We give a comparison principle in one dimension case.We consider the problem: The blow-up of the solutionsTheorem Assume u （ x ）∈C2.1 （QT ）∩C1.1（Q T ） is a plus solution of （A） - （C）. Assume exists a continuity function m（u）, 0≤u<∞, satisfies If u0 > 0, we have Then the solutions may blow-up at finite time.The boundedness of the solutionsTheorem Assume fora,b> 0, there exists aA, if t≥A, then and for , t≥A, we haveFor （ x , t ）∈[0, l]×[ 0,∞), the plus solutions of the problem （A）- （C） may boundedness at finite time.
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