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In this paper,we study the threshold result for Robin problem of the porous equation.We mainly focus on two classes of initial boundary value problems.The first one is the following initial-boundary value problem of homogeneous porous equation whereΩis a bounded domain in Rn,p/m∈(1,n+2/n-2),β>0. The steady-state problem of (1) is The important feature for (1) is that any two distinct solution of (2) must inter-sect somewhere.Using this feature,and combining a priori estimate for the global solution of (1) with sub-super solution method,we prove that any positive solution of (2) is an initial datum threshold for the existence and nonexistence of global solution to (1). More precisely,we haveTheorem A:Assume U(x) is an arbitrary solution of (2).Then the following conclusion hold(ⅰ) If 0≤u0(x)≤ηU(x), and 0<η<1, then problem (1) has a global solution u(x,t;u0). moreover, (?)u(x,t; u0)= 0.(ⅱ) If u0(x)≥ηU(x), andη> 1,then the solution u(x,t:u0) of problem (1) blows up in finite time. That is, there exists 0<Tmax<+∞, such that (?) sup u(x, t;u0)=+∞.The second one is the following initial-boundary value problem of inhomoge- neous porous equation whereΩis a bounded domain in R",0<β<+oo;p/m∈(1,n+2/n-2,f(x)∈C(Ω) f(x)≥0 and f(x)≠0. The steady-state problem of (3) is The difference between (1) and (3) is that any two distinct solution of (4) may not intersect. But the structure of solution set of (4) is very good. That is, there exists a positiveλ*, such that if 0<λ<λ*, there is a minimal solution Uλfor problem (4). Moreover, any two different solution which are distinct to Uλeither equivalent or intersect somewhere.By making use of this property and the same method as that in the proof of Theorem A, we can prove the following result.Theorem B:If 0<λ<λ,, and Uλ(x) is the minimal solution of problem (4). If uλis an arbitrary solution of problem (4) which is distinct to Uλ, then we have(i) If 0≤u0(x≤ηUλ(x). and 0<η<1, then there exist a global solution u(x,t;u0). Moveover. lim u(x.t:u0)=Uλ(x).(ii) If uo(x)≥ηUλ(x). andη>1. then the solution u(x,t;u0) blows up in finite time.
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