Cauchy Problem Of Semilinear Double Temperature Heat Equation

In this paper,we study the Cauchy problem for a class of semilinear double temperature heat equation u_t-â–³u-â–³u_t+u=f(u),x∈Rⁿ,t＞0, u(x,0)=u₀(x),x∈Rⁿ. Semilinear double temperature heat equation is one of nonlinear pseudoparabolic equations arised from physics.This paper adopts the method of potential well and the family of potential wells.Firstly,we study the invariant sets of solutions by using the family of potential wells and obtain vacuum isolating behavior of solutions.Secondly,we study the the existence of the global weak solutions and blow-up of solutions.We obtain a threshold result of global existence and nonexistence of solutions.Thirdly,we study the existence of the global weak solutions of the problem with critical initial condition J(u₀)=d by using the method of Galerkin and the family of potential wells.Finally,we study the asymptotic behavior of solutions of the problem.