Blow-up Properties Of Solutions To A Class Of Parabolic Type P-Kircchhoff Equations

The main purpose of this paper is to consider the following initial-boundary value prob-lem for a parabolic type p-Kirchhoff equation with general diffusion coefficient and general nonlinearity where?(?)Rn(n? 1)is a bounded domain with smooth boundary(?)?,T?(0,+?]is the maximal existence time of the solution u(x,t),p>2 and up? W01,p(?).The diffusion coefficient M(t),the nonlinearity f(s)and the time-dependent function k(t)are supposed to satisfy the following assumptions;(HI)M?C[0,?)and M(t)?m0>0 for some m0 and for all t? 0.Moreover,there exists a constant ??(0,1)such that M(t)??tM{t),(?)t?R+,where(?)(H2)sf(s)?0,(?)s?R;(H3)f?C1(R),and there exists a constant ?>p/?-1 such that s[sf'(s)-?f(s)]?0,(?)s?R;(H4)There exist a positive integer l and constants ai>0(1?i?l)such that where 1<p1<...<pl<2*-1,2*is the Sobolev conjugate of p,i.e.,p*=+? for n?p and p*=np/n-p for n>p;(H5)k?C1[0,?),k(0)>0 and k'(t)?0 for all t?[0,?).This paper is divided into four chapters.In the first chapter,we first introduce the phys-ical background and development status of Kirchhoff equation,and then describe the main research content considered and the methods used in this paper.In the second chapter,we introduce some necessary notations,definitions and lemmas.In the third chapter,firstly,by employing the first order differential inequality method,we prove that the solutions to prob-lem(0.1)blow up in finite time when the initial energy is negative.Secondly,by applying Levine's concavity argument,we establish the sufficient condition for the finite time blow-up solution to problem(0.1)with positive initial energy.Finally,we derive the lower bound of the blow-up time for the above two cases.In the fourth chapter,we consider a special case of problem(0.1),i.e where q?2p-1 and q<p*-1.We prove that 2p-1 is critical for the existence of finite time blow-up solutions to problem(0.2).W?={u?W01,p(?)|I?(U)>0,J(u)<d(?)}? {0},where(?),d(?)is the depth of the potential well W?.The main results as follows:Theorem 1.Any weak solution u(x,t)to problem(0.1)blows up at some finite time T provided one of the following statements holds:(?)J(u0;0)<0;(?)0?J(u0;0)<m0/Sp2(?/p-1/?+1)(?u0?22-1)?C0(?u0?22-1),C0>0 by(H1)and(H3).Moreover,an upper bound for T has the following form:When(?)holds,T??u0?22/(1-?2)J(u0;0);When(?)holds,T?4??u0?22/(?-1)2(?+1)[C0(?u0?22-1)-J(u0;0)]Theorem 2.Let all the assumptions in Theorem 1 hold,and assume that 1<pl<p+2p/n-1.Then the maximal existence time of problem(0.1)satisfies T??u0?22(1-h)/2(h-1)c'where h is a positive constant that depends on p,pl,n,the positive constant C depends on ai,pi(i=1,…,l),?,m0.Theorem 3.Any weak solution u(x,t)to problem(0.2)exists globally provided one of the following assumptions holds:(?)1<q<min{2p-1,p*-1};(?)q=2p-1<p*-1 and b?S2p2p.Moreover,when(i)holds,whereWhen(?)holds,Theorem 4.Assume that p>2,q=2p-1<p*-1,b<S2p2p and u0?W01,p(?).If J(u0)<d,I(u0)>0,then problem(0.2)admits a global solution u?L?(0,?;W01,p(?)),ut?L2(0,?;L2(?)),and u(t)?W for any 0?t<?.Moreover,?u?22?[?u0?22-2p+A*(p-1)t]-1/(p-1)where the constant A*=2b(1-?1)/S2p2p>0,?1<1 is a root of the equation d(?)=J(u0).Theorem 5.Assume q=2p-1<p*-1,b<S2p2p and let u(x.t)is a weak solution to problem(0.2),u0?W01,p(?).If J(u0)?d,I(u0)<0,then u(x,t)blows up at some finite time T.