| 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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