Global Existence, Uniform Decay, And Bow-up Of Solutions For A Singular Nonlocal Viscoelastic Problem

In recent years, more and more attentions of viscoelastic problems have been paid as the wide applications in fluid saturation, porous media, soil analysis and other related fields. Re-searchers have established a large number of related results concerning local existence, global existence, asymptotic and blow-up of solutions. This work deals with a singular nonlocal vis-coelastic problem with the Dirichlet and nonlocal integration mixed boundary condition.In Chapter2, we consider the local existence of solutions for a singular nonlocal viscoelastic problem based on [24] with a possible damping term. We construct some appropriate function spaces and establish a priori estimate of the strong solution. Based on the density arguments, we establish the solvability of the linear case. Then, by using the contraction mapping theorem and the solvability of the nonlinear case is proved.Since the solvability of the problem has established, Chapter3deals with the global exis-tence, blow-up and decay behaves of the problem. By using the potential well theory and the modified convexity method, we obtain that the solution blows up in finite time with the initial data in the unstable set. Also, we use the potential well theorem to establish the global existence of solution with the initial data in the stable set. The condition on the relaxation function is replaced by a more general form:g’(t)≤-ζ(t)gr(t)(1≤r<3/2), which is a generalization of [1,17,24,27,28]. Then, based on the perturbed energy method, we establish a more general energy decay.In Chapter4, we consider the problem without the nonlinear source term. By assuming that the relaxation function has some small flat zones and these zones are not too big (i.e. g’=0in some parts of the zone), and the flatness rate is bigger than that in [39], we establish an arbitrary decay result instead of just exponential or polynomial type no matter a=0or a>0. To achieve our goal, we construct some new functionals, use the perturbed energy method with some inequalities skills and make different choices for the cases a=0and a>0, respectively.Chapter5deals with a singular nonlocal viscoelastic problem with nonlinear damping term h(ut) and nonlinear source term bu|u|p-2. By constructing a Lyapunov functional and using the perturbed energy method, we establish a general decay rate result without imposing any restrictive growth assumption on the damping term.