We consider the initial-boundary value problem for finitely degenerate parabolic equation. We first give sufficient conditions for the blow-up and global existence of the parabolic equation at high initial energy level. Then, we establish the existence of solutions blowing up in finite time with initial data at arbitrary energy level. Finally, we estimate the upper bound of the blow-up time under certain conditions.
运用Mawhin重合度理论,讨论一类半直线上三阶多点边值问题{x″′(t)=f(t,x(t),x′(t),x″(t))+e(t),t∈[0,+∞),x(0)=m∑i=1α_(i)x(ξ_(i)),x(1)=n∑j=1β_(j)x(η_(j)),limt→+∞x′(t)=∑k=1lγkx″(ζk)在dim Ker L=3共振情形下解的存在性,f:[0,1]×R→R满足S-Carathéodory条件,e∈L1[0,∞),α_(i),β_(j),γk∈R,0<ξ1<ξ2<…<ξm<+∞,0<η1<η2<…<ηn<+∞,0<ζ_(1)<ζ_(2)<…<ζ_(l)<+∞(m,n,l∈Z^(+)),并且满足下列条件:(C1)m∑i=1α_(i)=1,m∑i=1α_(i)ξi=0,m∑i=1α_(i)ξ2i=0,∑j=1nβ_(j)=1,∑j=1nβ_(j)η_(j)=1,∑j=1nβ_(j)η2j=1,∑k=1lγk=1;(C2)Δ=∣∣∣∣Q_(1)e−tQ_(1)te−tQ_(1)t2e−tQ_(2)e−tQ_(2)te−tQ_(2)t2e−tQ_(3)e−tQ_(3)te−tQ_(3)t2e−t∣∣∣∣:=∣∣∣∣a_(11)a_(21)a_(31)a_(12)a_(22)a_(32)a_(13)a_(23)a_(33)∣∣∣∣≠0,其中,Q_(1)y=m∑i=1α_(i)∫ξi_(0)∫^(s)_(0)∫^(τ)_(0)y(v)dv dτds,Q_(2)y=∑j=1nβ_(j)∫^(η_(j))_(0)∫^(s)_(0)∫^(τ)_(0)y(v)dv dτds,Q_(3)y=l∑k=1γk∫^(+∞)_(γk)y(s)ds。
