Upper And Lower Bound Estimates Of Blow-up Time To A Class Of Parabolic Type Kirchhoff Equation With Arbitrary Initial Energy

In this paper we consider the following nonlinear parabolic type Kirchhoff equations,where the diffusion coefficient has the specific form M(s)= a + bs with positive parameters a,b,Ω(?)Rn(n ≥ 1)is a bounded domain with smooth boundary(?)Ω,3<q<2*-1,where 2*is the Sobolev conjugate of 2,i.e.2*=+∞ for n =1,2 and 2*=2n/n-2 for n ≥3.Under some assumptions,we study the blow-up problems of solutions and point out upper and lower bound estimates of blow-up time.Firstly,we introduce the background and development of nonlocal parabolic problems related to the problem in this paper.Secondly we construct some appropriate energy func-tionals J(u)and I(u),give the definitions of weak solution and blow-up time,some basic inequalities as well as the main conclusions of this paper.Finally,by using the concave method of Levine,we estimate the upper bound of blow-up time of solutions to the parabolic problems with nonnegative initial energy.Besides,by the negativity of the energy function-al I(u)and Gagliardo-Nirenberg inequality we obtain a first-order differential inequality.Moreover,we get the lower bound estimate of blow-up time through integrals as well as the definition of blow-up.The main conclusions of this paper are as follows:Theorem 1.Assume that 3<q<2*-1 and that u(x,t)is a weak solution to Problem(0.1),then the upper bound of the blow-up time T*has the following form:(i)If J(uO)<0,thenwhere λ1>0 is the first eigenvalue of-△ inΩ with homogeneous Dirichlet boundary condition.Theorem 2.Assume that all the assumptions in Theorem 1 hold and 3<q<1+8/n,then the maximal existence time satisfieswhereC is the positive constant in Gagliardo—Nirenberg inequality.