Study On The Properties Of Some Parabolic Equations

In Chapter 1,we investigate the following parabolic system with nonlinear non-local boundary condition where Ω is a bounded domain in RN(N≥1)with smooth boundary (?)Ω, and p,q,r>0. The initial datum u0(x),v0(x)∈C2+α(Ω)for some α∈(0,1), u0(x)≥0,υ0(x)≥0 in Ω and satisfy the compatibility conditions.f, g are nonnega-tive continuous functions defined for (x,y)∈(?)∈×Ω and satisfy (?)Ω f(x,y)dy> 0, (?)Ω g(x,y)dy> 0 for all x(?)Ω. We prove the following results:(i) Let p,q≤1. Then for all r≤1, the solution exists globally for any nonneg-ative initial data.(ii) Let p,q>1.Then if r>1 and h0|Ω|>Kmin{p,q},the solution blows up in finite time for large initial data.(iii) Let p>1>qorq>1>p. Then if pq<1,r≤1,and (?)Ω f(x, y)dy≤ 1,(?)Ωg(x,y)dy≤1 for all x∈(?)Ω,the solution exists globally for sufficiently small initial data.(iv) Let p>1>q or q>1>p. Then if pq>1,r≥1, and (?)Ωf(x, y)dy≥ 1,(?)Ωg(x,y)dy≥1 for all x∈(?)Ω, the solution blow up in finite time for sufficiently large initial data.In Chapter 2, we consider the following reaction-diffusion equation with nonlin-ear nonlocal boundary condition where p, q, l>0, Ω is a bounded domain in RN(N≥1)with smooth boundary (?)Ω, v is unit outward normal on (?)Ω. Here c(x,t) is a positive bounded Holder continuous function defined for x∈Ω, t∈[0,T] and k(x,y,t) is a positive bounded continuous functions defined for x ∈(?)Ω,y∈Ω,t>0. The initial datum u0(x) ∈C2+α(Ω) (?) C(Ω) with 0< a< 1, and further, we assume that u0(x)> 0,(?)0 and satisfies the compatibility conditions. We investigate the global existence and blow up in finite time of a nonnegative solution by using a sub-super solution method. We obtain the following results:(i) Assume that p+q≤1,l≤1.Then the solutions exist globally for any nonnegative initial data.(ⅱ) Assume that p+q<1,l>1.Then the solutions exist globally for small initial data.(ⅲ) Assume that p+q>1,l>0. Then the solutions blow up in finite time for any positive initial data.(ⅳ)Assume that min(p, q)>1,l>0. Then the solutions blow up in finite time for large initial data.In Chapter 3,we investigate a parabolic system where p, q> 0, Ω is a bounded domain in RN with a smooth boundary (?)Ω,λ>0 is a parameter, φ and φ are nonnegative continuous functions on Ω. We prove that the maximal existence time of blow-up solutions approaches the life span of the solution of a ODE system as A goes to oo.We prove the following results:Let p, q>0. Suppose φ,φ∈C(Ω) satisfy φ,φ≥0 in ,ψ=ψ=0 on (?)Ω, ψ+ψ(?)0.(i)If qM(?)>pM(?),then we have(ii)IfqM(?)<pMψ,then we haveIn Chapter 4, we consider the extinction properties of solutions for the following fast diffusion equation with a nonlinear source where Ω is an open bounded domain in RN with smooth boundary (?)Ω,0<m<1 and p,q>0.u0(x)L∞(Ω)(?)W01,1+m(Ω) is a nonnegative function. We establish the conditions for the extinction of solutions in finite time. It is interesting to observe that if p+q<m or p+q=m and aμm,q<λ1,the maximal solution is positive in Ω for all t> 0, if p+q>m, the solution vanishes in finite time for small initial data. We also give the decay estimates of solutions before the occurrence of the extinction are derived. Furthermore, for the sake of illustration, we also present a purely numerical example in which the solution u vanishes in finite time. Our results read as follows:(i) Assume that p+q<m. Then for any nonnegative initial datum u0, the maximal solution U(x,t) can not vanish in finite time.(ii) Assume that p+q=m and aμm>g>λ1. Then for any nonnegative initial datum u0, the maximal solution U(x, t) can not vanish in finite time.(iii) Assume that p+q>m.(a) If N> 2, then every nonnegative weak solution vanishes infinite time for small initial data uo(x). Furthermore, we have forN-2/N+2<m<1,and for 0<m<N-2/N.(b) If N=1,2, then every nonnegative weak solution vanishes infinite time for small initial data u0(x).Furthermore, we have(?)...