| This paper consists of four chapters. The first chapter is the introduction. In the second chapter,we will study the existence and uniqueness of the global generalized solution and the global classical solution to the initial boundary value problem for a class of linear evolutionary equation.In the third chapter ,we will prove the existence and uniqueness of the global generalized solution to the initial boundary value problem for a class of nonlinear evolutionary equation.In the fourth chapter,we will prove the blow up of the solution to the problem mentioned in chapter three.We will study the following nonlinear evolutionary equationwith the initial boundary value conditionsororwhere u(x,t) denotes an unknown function, k1,k2 are two positive constants, g(s),f(s) are given nonlinear functions, φ(x) and ψ(x) are given initial value function, ) denotes the gradient operator, Rn(n = 1,2,3) is a bounded domain with sufficiently smooth boundary , v is the outward normal to the boundary , , T > 0,QT = Ω×(0,T).We first consider the following linear equationwhere G(x,t)is a given function about x,t.We will prove the existence and uniqueness of the global generalized solution and the global classical solution to the three initialboundary value problems for the equation(8).Then by making use of the contraction mapping principle we prove the existence and uniqueness of the local generalized solution to the three initial boundary value problems for the equation(1).Theorem 1 Suppose v? G H4{￡l),xp G H2{V),G G C([0,T];L2(Q)),then there exists a unique global generalized solution u(x,t) to the problem (8),(2),(3)or the problem (8),(4),(5)or the problem (8),(6),(7)andu(x, t) G C([0, T];H4(n)), ut G C([0, T\;H2{Q)) n L2([0,T];tf4(ft)),utt G L2(Qt),and u(x,t)satisfies the identity/ I {utt-k1Au + k2A2utG(x,t)}h{x,t)dxdt = 0, VheL2(QT) Jo JnTheorem 2 Suppose y G #7(ft),V G H9{CL),G(x,t) G H\[0,T];H4(Q)) n C([0,r];/f4(^)),G(x,0) G //5(Q),V'G(o;,t) = 0, (x,t) G dfi x (O,T),(i= 1,2,3,4), then there exists a unique global classical solution u(x,t) to the problem (8),(2).(3)or the problem (8),(4),(5)or the problem (8),(6).(7).Theorem 3 Suppose ..,. 1{C8(1)[32S2(CC/)(1 + C42C/2 + C2lC42^4 + AC^U2) + 8p(CU)}}1}Then/? : Q{D\ T) -> Q(6r, T)is strictly contractive.Theorem 4 Under the conditions of theorem 3 , there exists a unique local generalized solution u(x,t) to the problem (l),(2),(3)or the problem (l).(4).(5)or the problem (1).(G).(7). u(x.t) G C([0.T0):Hl(Q)):andvt G C([0, To);i/2(fi))nl2([0. r0): //'(^)). utt G L2(Qt0)-where [0, 7o)is the maximal time interval.In chapter four,we will prove the solution of the problem (l),(2),(3)or the problem (l).(4).(5)or the problem (1).(6),(7)blows up in finite time by means of the concavity method.Theorem 5 Supposesg{s) < 2(2/3+ whereG(s) = /os g(r)dT , f3 > 0 is a constant.whereF(s) = ￡ f(r)dT(H4)E(Q) = |M|2 + h\\Vip\\2 + 2 /? G(& 0(B)E(0) = 0,2 /n 00,2/n<^dz + A;2||A9||2 > 2^2/5(2/?+lj^Oj^Hlkll2 + 2)...
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