The Investigation On Some Problems Of The Nonlinear Evolution Equations

The nonlinear evolution equations are put forward from a large number of practical problems such as mechanics,fluid mechanics,physics,chemistry,biology and population dynamics and so on.Therefore,the equations have important practical background and wide applications,and are one of the most important research directions on the theory of nonlinear partial differ-ential equations.No matter from theoretical aspect,or numerical calculation,studies on this kind of equations help us know the state or process evolving over time in the natural science.In this paper,we study the initial bound-ary value problem of a class of semilincar pscudo-parabolic equation with a nonlocal source and damped semilincar wave equation with logarithmic non-linearity,as well as bounds for blow-up time of a nonlinear viscoelastic wave equation and of a semilinear pseudo-parabolic equation with nonlocal source.The main results are as follows:·We study the initial boundary value problem of a class of semilinear pscudo-parabolic equation with a nonlocal source.First,by combining the Galerkin method and potential wells,we study the global existence of weak solutions and obtain corresponding corollaries in the cases of J(u0)<d and J(u0)= d,respectively.Furthermore,we give the exponential asymptotic behavior and the blow-up criterion of weak solutions,from which we get a threshold result of global existence and nonexistence of solutions.Then,we obtain the important comparison principle.The global existence of weak solution and abstract blow-up criterion are considered under the condition of J(u0)>d.·We study the initial boundary value problem of damped semilinear wave equation with logarithmic nonlinearity.First,by utilizing new method we in-troduce a family of potential wells,and obtain the existence of global solutions.Then,the invariance of the family of potential wells under the flow is re-searched and vacuum isolating property of the solutions is gained.Finally,we consider energy decay estimate by improved integral estimate method.The results indicate that the energy will decay to zero exponentially as t ? ?.·We study bounds for blow-up time of a nonlinear viscoelastic wave e-quation and of a semilinear pseudo-parabolic equation with nonlocal source.By using appropriate auxiliary function and first order differential inequalities as well as Sobolev type inequality,we obtain the upper and lower bounds for blow-up time of the above two kinds of equations.