Lower Bound For The Blow-up Time Of The Solution To A Viscoelastic Equation With Strong Damping And Nonlinear Source Term

This paper mainly considers the lower bounds for the blow-up time to a viscoelastic equation with strong damping and nonlinear source term.Our main problems studied is as follows(?)Owing to the presence of strong damping and viscoelastic terms,it brings us some d-ifficulties in studying the above problems.In this paper,we introduce the control function,apply Sobelev embedding inequalities and interpolation inequality to establish several class-es of differential inequalities,then obtain the estimates of lower bounds for blow-up time of solutions of the problem(0.1).As we well know,the source term may cause blow-up of solutions while the damping term may make the solution be stable.In fact,according to Lipschitz continuity of the mapping ?:|u|p-2u:H1(?)?L2(?),p will be divided into the following three cases:Subcritical:1<p<2N-2/N-2;Critical:P=2N-2/N-2;Supercritical:2N-2/N-2<p<N+2/N-2Super-supercritical(No definition of energy functional):N+2/N-2?p<2N/N-2.Define E(0)as the initial energy functional..For subcritical case,we have the following conclusion:Theorem 0.1.If the following conditions are satisfied,(1)?>0,?>-??1,?1is the first eigenvalue under the homogeneous initial boundary value condition,(2)2<p<2(N-1)/n-2(N?3),(3)g is C1 function and1-?0?g(s)ds=l>0,(4)u0 ?H01(?),u1 ?L2(?),and ??u0udx>0,E(0)?d,then the blow-up time T*satisfies the following estimate:(?)where C2=pC2q-1((1+1/pl))q,C3=2pE(0)+(2E(0)/lqpC1,q=p-3/2,C is a Poincare constant,H(0)H(0)=??|u0|pdx.Due to the failure of the embedding H1(?)?L2(?),we have to look for a new method to discuss the supercritical case.We apply interpolation inequality and Sobolev inequality to obtain an inverse Holder inequality,and then obtain the following result:Theorem 0.2.If conditions(1),(3)and(4)remain true and the following condition is satisfied,2(N2-2)/N(N-2)<p<2N/N-2(N?3),then the blow-up time T*of the solution to problem(0.1)satisfies the following estimates:(?)C32 = 2pl/pl-2|E(0)|+[2p/pl-2|E(0)|]N/N-2-1Ck,k = 2(N-1)/N-2.(?)Finally,we discuss the super-supercritical case 2(N2-2)/N(N-2)<p<2N/N-2(N?3),Our main result is as follows:Theorem 0.3.If Conditions(1),(3)and(4)remain true and the following condition is satisfied,2(N2-2)/N(N-2)<p<2N/N-2(N?3),then the blow-up time T*of the solution to problem(0.1)satisfies the following estimates:T*??H(0)?1/D1yq+D2dy,where D1=Cp?12p-3/2(C13)q,D2=(pC13)qEq(0),C13=?+?1?/p(?+?1?-??1,Cp1/?1 =?,??1/p(?+?1?)<1,q =p-1.