Research On The Critical Index And Blow-up Problem Of Solutions For Pseudo-parabolic Equations

The pseudo-parabolic equation is a kind of non-classical diffusion equation,which has a higher order term about the mixed derivatives of time-space,and is used to describe the mutual diffusion theory between viscous incompressible fluids,the seepage theory,the aggregation mechanism of endangered species,and the unidirectional propagation of dispersive long waves.It has become an important branch in the field of partial differential equations.Firstly,the Cauchy problem of pseudo-parabolic equation ut=Δu+kΔut+‖w(x)u‖Lq(Rn)p-1ur+1(x,t)with weighted sources is studied,and the complete critical Fujita index pc=1+(2q-nqr)/(nq-(n-qs)+)of the equation is obtained.We use the test function method and Kepler method to get the blasting behavior of the solution,and use the compression mapping principle to prove the existence of the whole understanding.Meanwhile,the non-local terms in the equation are treated by using Holder inequality and other methods.The results show that a smaller weight function w(x)is more conducive to the global existence of the solution,while a larger k in pseudo parabolic term kΔut is more conducive to the global existence of the solution.Next,the second critical index and life span of pseudo-parabolic equation ut-kΔut=Δu+w(x)up with non-homogeneous term are discussed.By introducing the equivalent(integral form)diffusion equation,the conclusion of solution blasting is obtained.At the same time,a suitable upper solution is constructed,the conclusion of the whole existence of the solution is obtained,and the second critical index a*=(2+σ)/(p-1)of the equation is given.The life span of blasting solutions is estimated by self-similar transformation and comparison principle.